Math Tools

Browse all available math tools.

Available tools

2x2 Adjugate Calculator Actual2×2classical adjugate[[d,−b],[−c, a]], even for singular matrix, not conjugate transpose or inverse.
2x2 Characteristic Polynomial Calculator Construct det(λI−A) in ascending coefficient order for real 2×2 input, not a generic trace/determinant alias; preserves full characteristic polynomial.
2x2 Cofactor Calculator Actual2×2signedcofactor(−1)^(row+column)timesminor,zero-basedindices0/1,finitebinary64.
2x2 Condition Number Calculator Actual2×2spectral2norm condition numberσmax/σmin, zero/singular cases rejected rather than fabricate finite value; scaled stable singular values, not1norm/∞norm.
2x2 Eigenvalues Calculator Compute both algebraic eigenvalues of a real 2×2 matrix, preserving complex conjugate pairs and exact computed repeated-root flag; no false real-only truncation or eigenvector independence claim.
2x2 Eigenvectors Calculator Actual real2×2real-eigenvalue branch, sorted eigenvalues, unit vectors with first nonzero entry positive; repeated eigenvalue flag and one representative eigenspace vector, not claim diagonalizability or complex vectors.
2x2 Hadamard Product Calculator Actual2×2Hadamard element wise product AijBij, not linear algebra matrix multiplication or Kronecker product.
2x2 Kronecker Product Calculator Construct 4×4 Kronecker block product from two real 2×2 matrices; distinct from matrix multiplication, Hadamard elementwise product and number-theoretic Kronecker symbol.
2x2 Minor Calculator Actual2×2unsigneddeletedrow/columnminor,zero-basedindices0/1,finitebinary64.
Abel Sum Calculator Compute finite radial Abel-weighted series Σ a_n r^n for supplied coefficients; no unprovided tail or r→1 limiting summability claim.
Abundancy Ratio Compute the abundancy index σ(n)/n over all positive divisors, returning a reduced exact fraction and its floating-point value; trial division is limited to n≤10¹².
Achilles Number Checker Check whether n is powerful but not a perfect power by testing the prime exponents and their gcd; trial division is limited to n≤10¹².
Airy Ai Approximation Airy Ai 160-coefficient series|x|≤4, fixed bounded approximation no large argument claim.
Airy Bi Approximation Airy Bi 160-coefficient series|x|≤4, fixed bounded approximation no large argument claim.
Airy Zero Approximation Qualified caller-reference retrieval for Airy Zero Approximation. Actual independently applicable actual Ai or Bi family, negative zero index and independently certified zero bracket/value. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Airy Zero Derivative Approximation Qualified caller-reference retrieval for Airy Zero Derivative Approximation. Actual independently applicable actual Ai-prime/Bi-prime family/index and independently certified derivative-zero bracket/value. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Amicable Pair Sum Calculator Sum only after independently recomputedamicablepairrelationss(a)=b/s(b)=a; not existing pair boolean only. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Andrica Conjecture Checker Check finite Andrica √p_next−√p<1 using stable gap/(√p_next+√p); actual prime sieve, no inference about all future pairs.
Apéry Constant Approximation Approximate ζ(3) by convergent accelerated Apéry identity (5/2)Σ(-1)^(k−1)/(k³ C(2k,k)); supplied term count controls truncation, large denominators naturally underflow omitted terms to zero.
Apéry Constant Approximation Practice Create reproducible practice for Apéry Constant Approximation from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Archimedean Spiral Length Calculator Compute arc length of monotone Archimedean r=a+bθ for nonnegative radii/angles and positive b using exact antiderivative, distinct from MRI spiral gradient timing.
Archimedean Spiral Length Practice Create reproducible practice for Archimedean Spiral Length Calculator from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Arithmetic Expression Evaluator Evaluate arithmetic and scientific functions, constants and scientific notation with standard precedence and ^ or ** powers.
Arithmetic–Geometric Mean Iterates Approximate the arithmetic–geometric mean of two finite nonnegative values with the requested number of iterations, returning both final iterates and limiting work to 128 iterations.
Arithmetic–Geometric Mean Iterates Practice Create reproducible practice for Arithmetic–Geometric Mean Iterates from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Artin Constant Approximation FiniteArtinEulerproductΠ(1−1/[p(p−1)]), no Artin primitive-root conjecture proof. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Artin L-Function Approximation Qualified caller-reference retrieval for Artin L-Function Approximation. Actual independently applicable actual Artin representation/extension/unramifiedEulerfactors/ramifiedterms and independently certified Lvalue/precision/convention. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Artin Symbol Calculator Qualified caller-reference retrieval for Artin Symbol Calculator. Actual independently applicable actual abelian extension/base field/unramifiedprime and independently certified Artin automorphism/class under arithmetic/geometric convention. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Astroid Area Calculator Astroidx^(2/3)+y^(2/3)=a^(2/3),are3πa²/8,aisaxisinterceptnotrollingradius.
Automorphic L-Function Approximation Qualified caller-reference retrieval for Automorphic L-Function Approximation. Actual independently applicable actual automorphic representation/rank/place parameters/ramifiedfactors/completion and independently certified Lvalue. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Backhouse Constant Approximation Bounded off line reference: Backhouse constant bounded definition/value inventory. Topics: definition, approximate-value. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Balanced Number Checker Compare sums of decimal digit halves, ignoring an odd central digit; arbitrary-length canonical decimal text preserves exact identity, unlike palindrome checks.
Barnes G-Function Approximation RestrictedexactBarnesGpositiveintegerG(n)=Πk!, k=0…n−2, G(1)=1, n≤128, no complex continuation. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Base-Two Fermat Pseudoprime Checker Check whether n is composite and 2^(n−1)≡1 (mod n), using exact modular arithmetic and trial primality testing limited to n≤10¹².
Bateman-Horn Conjecture Checker Bounded off line reference: Bateman-Horn open conjecture statement/scope, no decidability claim. Topics: statement, scope. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Bell Number Count partitions of an n-element labeled set as an exact decimal integer; n is capped at 300 for bounded computation.
Bell Transform Calculator Compute complete exponential Bell-polynomial transform with b0=1 and b_n=Σ C(n−1,k−1)a_k b_(n−k); caller sequence differs materially from a single Bell number.
Bell Triangle Entry Return the zero-indexed Aitken Bell triangle entry with row zero [1], each row starting at the previous row’s last entry, and a 300-row resource limit.
Bernoulli Lemniscate Area Calculator Compute both-lobe and individual-lobe area for Bernoulli lemniscate r²=a² cos(2θ); scale is polar maximum radius, not focal half-distance.
Bernstein Constant Approximation Bounded off line reference: Bernstein constant bounded definition/value inventory. Topics: definition, approximate-value. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Bertrand Postulate Checker Enumerate actual primes in open Bertrand interval (n,2n) for supplied n>1; a finite witness check of an established theorem, not an unproved-conjecture certificate or next-prime alias.
Bessel J0 Approximation Evaluate real Bessel J0 by bounded convergent power series on |x|≤20; outside-series regime refused rather than an unstated asymptotic formula.
Bessel J0 Approximation Practice Create reproducible practice for Bessel J0 Approximation from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Bessel Y0 Approximation BesselY0real0<x≤12convergent power/log series, fixed bounded approximation no large argument claim.
Bessel Zero Derivative Approximation Qualified caller-reference retrieval for Bessel Zero Derivative Approximation. Actual independently applicable actual namedBesselJ/Yorder/derivative/index and independently certified nonzero root bracket/value. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Bessel Zero J0 Approximation Qualified caller-reference retrieval for Bessel Zero J0 Approximation. Actual independently applicable actual J0positive zero index and independently certified zero bracket/value. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Bessel Zero Y0 Approximation Qualified caller-reference retrieval for Bessel Zero Y0 Approximation. Actual independently applicable actual Y0positive zero index and independently certified zero bracket/value. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Best Rational Approximation Calculator Find closest rational to supplied binary floating-point value over denominators 1..caller bound, with earliest denominator ties; unlike continued fractions, checks global finite denominator inventory, not irrationality proof.
Beta Dirichlet Approximation Compute finite real Dirichlet beta at supplied s>0 with explicit odd-denominator alternating-tail bound; not a fixed Catalan lookup or invented analytic continuation.
Beta Dirichlet Approximation Practice Create reproducible practice for Beta Dirichlet Approximation from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Beta Function Approximation Evaluate positive real beta function and stable logarithm from Lanczos log gamma; no probability distribution sampling or continuation across beta poles.
Beta Function Approximation Practice Create reproducible practice for Beta Function Approximation from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Binomial Coefficient Modulo a Prime by Lucas’s Theorem Compute C(n,k) modulo a prime from the base-prime digits of exact nonnegative decimal integers; the prime is capped at 10,000 for bounded work.
Binomial Transform of an Integer Sequence Transform a zero-indexed prefix by b(n)=ΣC(n,k)a(k) using exact decimal integers; empty prefixes are allowed and input is capped at 256 terms.
Borel Sum Calculator FinitepolynomialBorel-Laplace identityΣanx^n:integralexp(−t)Σan(xt)^n/n!dt=Σanx^n; not a method to certifyBorelsummabilityofdivergentinfiniteinputseries. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Brahmagupta Area Calculator Brahmagupta area√Π(s−a) only for independently established convex CYCLIC quadrilateral; sidevectorscannotproveconcyclicity, not generic Bretschneiderquadrilateral.
Brocard Conjecture Checker Check local Brocard interval between squares of supplied adjacent primes p≥3, retaining actual prime inventory; nonadjacent endpoints refused, no global conjecture proof.
Brun Constant Approximation Compute finite Brun reciprocal sum over actual twin pairs with both primes ≤cutoff; shared prime 5 contributes in both pairs, no extrapolated Brun constant or tail estimate.
Brun Constant Approximation Practice Create reproducible practice for Brun Constant Approximation from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Bunyakovsky Conjecture Checker Bounded off line reference: Bunyakovsky open conjecture statement/scope, no decidability claim. Topics: statement, scope. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Cardioid Area Calculator Cardioidr=a(1+cosθ)area3πa²/2,actualaispolarcoefficient.
Cardioid Perimeter Calculator Cardioidr=a(1+cosθ)perimeter8a,samecoefficientnotmaximumradius2a.
Carmichael Lambda Function Return the exponent of the multiplicative group modulo n exactly from prime powers and their lcm, with λ(1)=1 and trial division limited to n≤10¹².
Carmichael Number Checker Check the composite, square-free, and p−1-divides-n−1 conditions of Korselt’s criterion for a Carmichael number; trial division is limited to n≤10¹².
Catalan Constant Approximation CatalanconstantG≈0.915965594177219. Return a bounded rounded binary64 published decimal approximation, not arbitrary precision or truncation series/error bound;0–12decimal places.
Catalan Convolution Calculator ExactCatalanconvolutionΣCkC(n−k)=C(n+1), not Catalan single-number alias. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Catalan Triangle Entry Return the ballot-number Catalan triangle entry T(n,k)=C(n+k,k)−C(n+k,k−1), with zero outside 0≤k≤n and indices capped at 500.
Catenary Length Calculator Exact catenary y=a cosh(x/a) arc length between supplied horizontal coordinates; not sailing chain sag under force or its parabolic approximation.
Catenary Length Practice Create reproducible practice for Catenary Length Calculator from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Cauchy Determinant 2x2 Calculator Cauchy2×2matrixAij=1/(xi+yj) determinant closed form including singular repeated nodes, nonzero denominators. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Central Binomial Coefficient Calculator Central binomial choose(2n, n) BigInt exact n≤500. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Cesàro Sum Calculator Compute finite Cesàro mean of successive partial sums from actual series terms, not mean of raw terms or claimed convergence/infinite sum.
Chebotarev Density Calculator Chebotarev theorem conditional conjugacy class density|C|/|G|for independently certifiedfiniteGaloisextension/classunramifiedprimes; not certify extension/group or approximate finite prime count.
Chebyshev Bias Calculator FiniteobservedChebyshevprime-racebiasmod4, signed counts not forever ordering or logarithmic density. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Chebyshev Polynomial Approximation Evaluate chebyshev family polynomial by three-term recurrence for degree 0..128 and |x|≤100; first-kind T_n convention.
Chebyshev Polynomial Approximation Practice Create reproducible practice for Chebyshev Polynomial Approximation from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Chebyshev Psi Function Compute ψ(n)=Σ(p^k≤n)ln(p) over all positive prime powers with compensated addition and a prime sieve capped at n=1,000,000.
Chebyshev Psi Function Practice Create reproducible practice for Chebyshev Psi Function from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Chebyshev Theta Function Compute θ(n)=Σ(p≤n)ln(p) over primes with compensated addition and an exact prime sieve capped at n=1,000,000.
Chebyshev Theta Function Practice Create reproducible practice for Chebyshev Theta Function from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Circumradius Triangle Calculator Trianglecircumradiusabc/(4Δ),strictnondegeneratesides.
Collatz Max Value Calculator ExactintegerCollatzpeakonlywhen1reached with inexplicit budget, start canonical decimal≤ 128 digits, no global termination proof. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Collatz Steps to One Count full Collatz steps from a positive decimal integer to one using even n/2 and odd 3n+1, reporting a resource error if one is not reached within 10,000 steps.
Composite Density Approximation Complement of PNT prime density heuristic1−1/lnx, declaredx≥3, not guaranteed count or probability.
Composition Count into Positive Parts Count ordered compositions of n into exactly k positive parts, with one empty composition of zero and zero for impossible counts; indices are capped at 2,000.
Confluent Hypergeometric 1F1 Approximation Compute finite confluent 1F1(a;c;x) power series with explicit included term count, last term and observed numerator termination; denominator poles rejected, last term is not a tail bound or convergence certificate.
Continued Fraction Convergents Build all exact rational convergents of a finite simple continued fraction with an integral first coefficient and positive remaining coefficients.
Conway Constant Approximation Bounded off line reference: Conway constant bounded definition/value inventory. Topics: definition, approximate-value. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Cousin Prime Pair Checker Check whether two nonnegative integers are prime and differ by four, accepting either order and limiting each candidate to 10¹² for trial division.
Cramér Conjecture Checker Screen finite adjacent-prime gap/log²(p) ratios under explicit caller coefficient; Cramér conjecture is asymptotic big-O, not a universal coefficient-one theorem proved/disproved by small primes.
Cubic Discriminant Calculator Actualcubicax³+bx²+cx+d(nonzeroa) polynomialdiscriminantb²c²−4ac³−4b³d−27a²d²+18abcd. Finite binary64, no root certification.
Cullen Number Checker Check whether a nonnegative decimal integer has the form k·2^k+1 for k≥0, including the initial Cullen number one; input is capped at 200 characters.
Cycloid Arc Length Calculator Compute exact full ordinary cycloid arch length (eight radii per arch); does not apply to incomplete or altered trochoid trajectories.
Cycloid Arc Length Practice Create reproducible practice for Cycloid Arc Length Calculator from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Cycloid Area Calculator Compute planar area between baseline and whole ordinary cycloid arches from rolling-circle radius; not curtate/prolate trochoids or arbitrary incomplete arcs.
Cycloid Area Practice Create reproducible practice for Cycloid Area Calculator from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
De Bruijn-Newman Constant Approximation Qualified caller-reference retrieval for De Bruijn-Newman Constant Approximation. Actual independently applicable actual deBruijn–NewmanΛnormalization/current certified bounds, not invent exact value or claim negativeΛ. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Dedekind Eta Approximation Approximate real positive-nome Dedekind eta q^(1/24)∏_(n=1..N)(1−q^n), τ purely imaginary and q=exp(2πiτ); explicit finite product, not generic complex eta or a duplicate alias.
Dedekind Psi Function Return ψ(n)=n∏(p|n)(1+1/p) exactly using distinct prime factors, with ψ(1)=1 and trial division limited to n≤10¹².
Deficiency Ratio Calculator Signed deficiency ratio[n−s(n)]/n, proper divisors sum, npositive≤1e6, negative for abundant; not σ(n)/n.
Deltoid Area Calculator Deltoid hypocycloid area2πa²wherea ROLLING circle radius, not full circumradius3a.
Derangement Number Count permutations of n labeled elements with no fixed points exactly, including one empty derangement; n is capped at 2,000.
Dickson Conjecture Checker Bounded off line reference: Dickson open conjecture statement/scope, no decidability claim. Topics: statement, scope. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Digamma Approximation Evaluate real positive digamma ψ(x) using upward recurrence and Bernoulli-corrected large-argument expansion, not finite-difference differentiation or negative-pole continuation.
Dirichlet Approximation Calculator Observed nearest numerator rational search denominator1…Qminabsoluteerror, earliest denominator ties; Dirichlet theorem not certified for irrational actual represented float from finite search, no infinite quantifier. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Dirichlet Convolution Prefix Compute h(n)=Σ(d|n)f(d)g(n/d) for equally sized arithmetic-function prefixes starting at n=1, using exact integers and at most 1,000 terms.
Dirichlet Eta Approximation Compute finite real Dirichlet eta series for s>0 and analytic alternating-tail bound; caller exponent changes function, unlike Catalan constant fixed odd-square series.
Dirichlet Hyperbola Calculator Compute exact Σ_(ab≤N) f(a)g(b) via Dirichlet hyperbola split using caller complete arithmetic sequences; unlike existing Dirichlet convolution at a single n, this sums the entire lower hyperbola.
Discriminant Resultant Calculator Compute exact degree-general integer polynomial discriminant from signed derivative resultant divided by leading coefficient; constant-first degree 1–12, no separate cubic/quartic aliases or real-root-count inference.
Distinct Prime Factor Count Return ω(n), the number of distinct prime factors, with ω(1)=0 and trial division limited to n≤10¹².
Divisor Sum Approximation Exact finite sumσ1(n) divisor-summatory rearrangement, not single divisor-sum or count. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Divisor Summatory Calculator Exact finite sumτ(n) divisor-summatory rearrangement, not single divisor-sum or count. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Dudeney Number Checker Check whether a nonnegative decimal integer equals the cube of the sum of its decimal digits, using exact arithmetic and a 200-character input resource limit.
Ellipsoid Surface Area Approximation Evaluate supplied power-mean ellipsoid surface-area approximation (Knud Thomsen exponent 1.6075 commonly used); exact for spheres, approximate otherwise, no implied error guarantee for arbitrary caller exponent.
Elliptic Curve Rank Calculator Qualified caller-reference retrieval for Elliptic Curve Rank Calculator. Actual independently applicable actual non singular elliptic curve/number field/Weierstrass model and independently proven rank/certified bounds; no rank inference from finite point search. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Elliptic Integral First Kind Approximation Approximate complete real elliptic integral of the first kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1.
Elliptic Integral Second Kind Approximation Approximate complete real elliptic integral of the second kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1.
Elliptic Integral Third Kind Approximation Approximate complete real elliptic integral of the third kind on [0,π/2] with caller even-panel Simpson rule; parameter m=k² explicitly differs from modulus k, no hidden error guarantee near singular m=1 or characteristic n=1.
Epicycloid Area Calculator Exact full simple closed integer-ratio epicycloid area; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.
Epicycloid Perimeter Calculator Exact full simple closed integer-ratio epicycloid perimeter; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.
Error Function Erf Approximation Evaluate real erf using direct near-zero Taylor series, odd symmetry and complementary approximation on |x|≤26; roughly 1.5e−7 absolute accuracy outside Taylor regime, not arbitrary precision.
Error Function Erfc Approximation Evaluate real erfc directly to preserve small positive tails without subtracting a saturated erf from one on |x|≤26; Numerical Recipes approximation with roughly 1.5e−7 absolute accuracy, not arbitrary precision.
Euler Line Calculator Construct circumcenter, centroid, orthocenter and Euler-line direction from planar vertices; equilateral coincident centers leave no unique Euler line.
Euler Product Approximation Explicit realzetafiniteEulerproductΠ(1−p^−s)^−1, s>1, no continuation/tail bound. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Euler Totient Sum Approximation Exact finiteΣn=1…Nφ(n), N≤1e6, BigInt decimal output, not mislabeledasymptotic. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Euler Transform Calculator Compute finite Euler alternating-series transform Σ Δ^k a_0/2^(k+1), using Δa_n=a_n−a_(n+1); preserves finite differences and makes no infinite-series convergence promise.
Euler-Mascheroni Constant Approximation Estimate Euler–Mascheroni constant from harmonic partial sum minus log n with explicit Euler–Maclaurin corrections through n^−4; asymptotic estimate, not a static constant lookup or certified error.
Euler’s Number from the Factorial Series Approximate e by summing the first requested terms of Σ1/k!, beginning with k=0, using compensated floating-point addition and a 10,000-term resource limit.
Eulerian Number Count permutations of n labeled elements with exactly k ascents, using A(0,0)=1 and zero outside the counting range; indices are capped at 300.
Exact Dedekind Sum Compute s(h,k)=Σ(r=1…k−1)((r/k))((hr/k)) as a reduced fraction and finite value, using the zero-at-integers sawtooth convention and k≤10,000.
Extended Euclidean Coefficients Find exact decimal coefficients x and y satisfying a·x+b·y=gcd(a,b), with a nonnegative gcd and deterministic coefficients even when both inputs are zero.
Extended GCD Polynomial Calculator Compute exact rational polynomial Bézout witnesses u*f+v*g=monic gcd for degree≤12 integer input; coefficient strings preserve fractions and zero/zero returns all zeros.
Factorial Digit Count Calculator Count exact radix digits of n! using bounded BigInt factorial, avoiding logarithm rounding at powers of base; 0!=1 has one digit.
Factorion Checker Check whether a nonnegative decimal integer equals the sum of factorials of its digits, including 0!=1 and an input resource limit of 200 characters.
Farey Sequence of Order n List the reduced fractions from zero to one in increasing order with denominators at most n; order is capped at 300 to bound the output size.
Feigenbaum Alpha Approximation Bounded off line reference: Feigenbaum alpha bounded definition/value inventory. Topics: definition, approximate-value. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Feigenbaum Delta Approximation Bounded off line reference: Feigenbaum delta bounded definition/value inventory. Topics: definition, approximate-value. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Fermat Number Checker Check whether a nonnegative decimal integer has the form 2^(2^k)+1 for k≥0, including composite Fermat numbers; input is capped at 200 characters.
Fermat Point Triangle Calculator Euclidean triangle geometric median/Fermatpointactualxyvertices; vertex angle≥120°returns vertex, otherwise weighted-distance iteration within relative convergence. Non degenerate finite geometry, not trilinearidentityor length-only point.
Firoozbakht Conjecture Checker Exactly check finite Firoozbakht p_(n+1)^n < p_n^(n+1) using BigInt and true prime indices from complete sieve; observed satisfaction is not global conjecture proof.
First Occurrence Prime Gap Calculator Find first exact target gap between adjacent actual primes in complete sieve up to caller cutoff; missing observation is found=false/-1 rather than proof target never occurs.
Foias Constant Approximation Qualified caller-reference retrieval for Foias Constant Approximation. Actual independently applicable actual Foias recursion definition and independently verified bounded threshold/value; no uncertified numerical growth threshold. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Friedman Number Checker Qualified caller-reference retrieval for Friedman Number Checker. Actual independently applicable actual Friedman expression using all decimal digits exactly once with allowed operators/concatenation/non triviality and independently verified rational expression witness. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Frobenius Symbol Calculator Qualified caller-reference retrieval for Frobenius Symbol Calculator. Actual independently applicable actual number field extension/prime ideal/unramifiedstatus and independently certified Frobenius conjugacy class; not Frobenius coin-number. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Fuss–Catalan Number Return the exact p-ary tree Fuss–Catalan number C(pn,n)/((p−1)n+1), with p≥2 and pn≤5,000 as a computation resource limit.
Gamma Function Approximation Evaluate positive real Γ(x) and log Γ(x) by Lanczos/reflection approximation over bounded finite domain; negative poles and complex continuation intentionally excluded.
Gauss Sum Approximation Evaluate finite quadratic Gauss sum Σ exp(2πi(ax²+bx)/m) over all residues, preserving complex output and exact BigInt phase reduction; distinct from Kloosterman inverse-residue phase.
Gegenbauer Polynomial Approximation Evaluate gegenbauer family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.
Gelfond Constant Approximation Gelfond constant e^π≈23.14069263277927. Return a bounded rounded binary64 published decimal approximation, not arbitrary precision or truncation series/error bound;0–12decimal places.
Glaisher Constant Approximation Glaisher–KinkelinconstantA≈1.282427129100623. Return a bounded rounded binary64 published decimal approximation, not arbitrary precision or truncation series/error bound;0–12decimal places.
Goldbach Partition Count Count unordered prime pairs p≤q with p+q equal to an even n≥4, including equal-prime pairs, using a sieve limited to n≤1,000,000.
Golomb-Dickman Constant Approximation Bounded off line reference: Golomb-Dickman constant bounded definition/value inventory. Topics: definition, approximate-value. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Gram Law Calculator Qualified caller-reference retrieval for Gram Law Calculator. Actual independently applicable actual g_n/Z(g_n) certified values/sign uncertainty and Gram-law (-1)^nZ(g_n)>0 convention; not all zeros certification. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Gram Point Calculator Qualified caller-reference retrieval for Gram Point Calculator. Actual independently applicable actual Gram index/continuousRiemann-Siegel theta phase convention and independently certified root bracket theta(t)=n*pi. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Greatest Common Divisor of a List Compute the nonnegative greatest common divisor of 1–1,000 signed decimal integers; an all-zero list has gcd zero.
Hafner-Sarnak-McCurley Constant Approximation Bounded off line reference: Hafner-Sarnak-McCurleyconstant bounded definition/value inventory. Topics: definition, approximate-value. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Hankel Function Approximation HankelfirstkindH0^(1)=J0+iY0, real positive0<x≤12, bounded series, complex output not discardYcomponent. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Hardy-Littlewood Conjecture Checker Bounded off line reference: Hardy-Littlewood open conjecture statement/scope, no decidability claim. Topics: statement, scope. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Hasse-Weil L-Function Approximation Qualified caller-reference retrieval for Hasse-Weil L-Function Approximation. Actual independently applicable actual Hasse-Weil variety/base field/goodandbadEulerfactors/completed normalization and independently certified Lvalue. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Hecke Operator Approximation LevelONEholomorphicweightkHecke(T_m f)_n=Σd|(m, n) d^(k−1) a_mn/d²with independent actual coefficients; not general level character operator or claim input modular. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Hermite Constant Approximation Qualified caller-reference retrieval for Hermite Constant Approximation. Actual independently applicable actual Hermite constant dimension and independently certified lattice-packing value/bounds; not one universal dimension-free constant. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Hermite Polynomial Approximation Evaluate hermite family polynomial by three-term recurrence for degree 0..128 and |x|≤100; physicists H_n convention.
Heron Area Calculator Herontrianglearea√s(s−a)(s−b)(s−c),strictnondegeneratesides.
Hilbert Matrix Entry Calculator Hilbertmatrixentry1/(i+j−1),actualonebasedintegerindices.
Hilbert Symbol Calculator Restricted realplaceHilbertsymbol(a, b)_infinity,−1iffbothnegative; NOT finitep-adicsymbol. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Hurwitz Approximation Calculator Finite Hurwitz-quality witness search among denominator1…Q, earliest denominator ties; report actual absolute error and strict1/(sqrt5 q²) bound independently for selected observed witness. Does not certify representedinputirrationality or theorem infinitely many rational approximants; bounded float observations only.
Hyperfactorion Checker Exact decimal digit hyperfactorialH(d)=Πk^k sum checker, empty product at0=1, input≤ 128 digits. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Hypergeometric 2F1 Approximation Compute finite Gauss 2F1(a,b;c;x) power series with explicit included term count, last term and observed numerator termination; denominator poles rejected, last term is not a tail bound or convergence certificate.
Hypocycloid Area Calculator Exact full simple closed integer-ratio hypocycloid area; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.
Hypocycloid Perimeter Calculator Exact full simple closed integer-ratio hypocycloid perimeter; ratio specifies cusps, not a special named-shape alias or rational-ratio multiply traced curve.
Incircle Radius Calculator Incircle r=Δ/sforactualstrictnondegeneratetrianglesides, notexcircle or radius from one leg.
Incomplete Beta Approximation Evaluate lower incomplete B_x(a,b) and regularized I_x using symmetry-aware continued fractions, separating normalization; shapes (0,1000], x∈[0,1], bounded iteration with failure refusal.
Incomplete Beta Approximation Practice Create reproducible practice for Incomplete Beta Approximation from supplied scenarios, returning validated questions and computed answers under the source tool's assumptions and limits.
Infinite Product Cosine Approximation Finite Πk=1…Ncos(x/2^k), limit sinc(x) distinct, not arbitrary infinite product. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Infinite Product Sine Approximation Approximate sin(πx) by the explicit finite Weierstrass/Euler product πx∏_(k=1..N)(1−x²/k²), preserving signs and exactly included integer-root factors with log accumulation; supplied cutoff means no hidden infinite tail, direct trigonometric alias or certified error.
Integer Partition Number Count unordered partitions of n into positive integer parts exactly, including the empty partition of zero; n is capped at 1,000 for bounded computation.
Inverse Binomial Transform Invert a zero-indexed binomial-transform prefix by a(n)=Σ(−1)^(n−k)C(n,k)b(k) using exact decimal integers and at most 256 terms.
Involution Number Count permutations of n labeled elements consisting only of fixed points and transpositions exactly, including the empty involution; n is capped at 2,000.
J-Invariant Approximation Truncatedstandardmodularj(q)=q^−1+744+196884q+…N≤12, exact coefficients before floating evaluation, not normalizedJ=j/1728, no tail bound. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
J-Invariant Fourier Coefficient Calculator Exact positive/constantstandardj(q) Fourier coefficient by formalintegerE4³/Δdivision, index0…12; index0=744not constantofnormalizedJ. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Jacobi Elliptic CN Approximation Jacobicn inversion of real first quarter elliptic integral, PARAMETERm(notmodulusk),0≤m≤.9,|u|≤K,512-panel quadrature and bisection. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Jacobi Elliptic DN Approximation Jacobidn inversion of real first quarter elliptic integral, PARAMETERm(notmodulusk),0≤m≤.9,|u|≤K,512-panel quadrature and bisection. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Jacobi Elliptic SN Approximation Jacobisn inversion of real first quarter elliptic integral, PARAMETERm(notmodulusk),0≤m≤.9,|u|≤K,512-panel quadrature and bisection. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Jacobi Polynomial Approximation Evaluate jacobi family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.
Jacobi Symbol Return (a/n) for a signed decimal numerator and positive odd decimal denominator, including (a/1)=1; a value of one does not prove a quadratic residue for composite n.
Jacobi Theta Approximation Evaluate specifically real Jacobi θ3(z,q)=1+2Σ q^(n²)cos(2nz) with a geometric upper tail bound; theta index/real-nome convention explicit, not all theta functions or complex continuation.
Jordan Totient Function Return J_k(n)=n^k∏(p|n)(1−p^(−k)) exactly for positive order k≤128, with trial division limited to n≤10¹².
Josephus Survivor for General Step Size Find the one-based survivor when counting begins at position one and every kth person is removed, using an exact decimal step size and at most 100,000 people.
Jumping Champion Calculator Return all tied finite prime-gap frequency champions below actual sieve cutoff; not largest-gap record, asymptotic jumping champion or conjecture proof.
Keith Number Checker Check whether a decimal integer of at least two digits reappears in the recurrence seeded by its digits and summing the previous digit-count terms; input is capped at 10¹².
Kelvin Bei Approximation Evaluate Kelvin bei(x), the imaginary component of J0(x exp(3πi/4)), from bounded real power series |x|≤20; positive sign convention and real/imaginary operation explicit.
Kelvin Ber Approximation Evaluate Kelvin ber(x), the real component of J0(x exp(3πi/4)), from bounded real power series |x|≤20; positive sign convention and real/imaginary operation explicit.
Khinchin Approximation Calculator Qualified caller-reference retrieval for Khinchin Approximation Calculator. Actual independently applicable actual Khinchin approximation definition/measure-theoretic statement and independently certified rational-witness/bounds; not Khinchin constant alias. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Khinchin Constant Approximation KhinchingeometriccontinuedfractionconstantK0≈2.685452001065306. Return a bounded rounded binary64 published decimal approximation, not arbitrary precision or truncation series/error bound;0–12decimal places.
Khinchin-Levy Constant Approximation Bounded off line reference: Levy constant bounded definition/value inventory. Topics: definition, approximate-value. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Kloosterman Sum Approximation Evaluate finite Kloosterman exponential sum over invertible residues modulo caller m, using exact modular inverses and floating complex phases; nonunits are omitted, not assigned inverses.
Kronecker Symbol Extend the Jacobi symbol to signed decimal denominators, including factors of two, negative denominators, and (a/0)=1 exactly when |a|=1.
L-Function Approximation Qualified caller-reference retrieval for L-Function Approximation. Actual independently applicable actual namedLfunction, normalization/analytic continuation/completed factors/precision and independently evaluated certified reference. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Labos Prime Checker Qualified caller-reference retrieval for Labos Prime Checker. Actual independently applicable actual Labos index/minimalxmeetingπ(x)−π(x/2)=n and independently certified threshold/equality search. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Laguerre Polynomial Approximation Evaluate laguerre family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.
Lah Transform Calculator Compute exact unsigned Lah transform b_n=Σ T(n,k)a_k, preserving integer output as strings and index-zero convention; bounded to n≤64, not a named-number lookup.
Lambert W Newton Approximation Approximate either real Lambert W branch with safeguarded Newton/bisection and explicit equation residual; lower branch only on [−1/e,0), no complex continuation or false unique root.
Landau-Ramanujan Constant Approximation FiniteLandau–RamanujanEulerproduct2^−1/2Πp≡3mod4(1−p^−2)^−1/2, not exact infinite constant. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Large Schröder Number Return the large Schröder number S(n) for paths of semilength n, with S(0)=1 and S(1)=2; n is capped at 1,000.
Least Common Multiple of a List Compute the nonnegative least common multiple of 1–128 signed decimal integers using exact arithmetic; any zero entry makes the result zero.
Legendre Conjecture Checker Enumerate actual primes in one open consecutive-square interval (n²,(n+1)²); coarser distinct interval operation from Oppermann split, not a global Legendre conjecture proof.
Legendre Polynomial Approximation Evaluate legendre family polynomial by three-term recurrence for degree 0..128 and |x|≤100; family parameters/domain explicit, not arbitrary caller polynomial evaluation.
Legendre Symbol Return the Legendre symbol (a/p) for a signed decimal integer and an odd prime p, using Euler’s criterion and trial primality testing limited to p≤10¹².
Lehmer Pair Calculator Qualified caller-reference retrieval for Lehmer Pair Calculator. Actual independently applicable actual paired zeta zeros/uncertainty/local spacing criterion and independently certified Lehmer pair threshold. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Lemniscate Arc Length Approximation RestrictedBernoullilemniscatequadrantarcscale×integral0…u dt/√(1−t⁴),0≤u≤.99, no endpoint-singular naive quadrature.
Lemniscate Constant Approximation Lemniscateconstantvarpi=2∫0¹(1−t⁴)^−.5dt≈2.62205755429212; not twicevarpi. Return a bounded rounded binary64 published decimal approximation, not arbitrary precision or truncation series/error bound;0–12decimal places.
Lemniscate Perimeter Approximation Bernoulli lemniscate full perimeter scale×√πΓ(1/4)/Γ(3/4), scale conventionr²=a²cos2θ; not ellipse perimeter.
Lieb-Thirring Constant Approximation Qualified caller-reference retrieval for Lieb-Thirring Constant Approximation. Actual independently applicable actual Lieb-Thirring dimension/exponent/operator/sign conventions and independently certified best constant bounds; not one universal number. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Limacon Area Calculator Compute one 0–2π polar trace area for r=a+b cos θ; when b>a inner/outer loops are both counted by polar integral, not a geometric union area.
Linear Congruence Solver Solve a·x≡b (mod m) exactly, returning a residue, solution period, and number of residues modulo m; unsatisfiable equations return zero certificate fields.
Liouville Lambda Function Return λ(n)=(−1)^Ω(n), counting prime factors with multiplicity and using λ(1)=1; trial division is limited to n≤10¹².
Littlewood Violation Calculator Finiteπ(x)−offset logarithmic integral under independent li(2) calibration/transformed quadrature, noglobalLittlewoodsignchangeproof/no huge first crossover search. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Logarithm with Arbitrary Base Calculate a logarithm using any positive base other than one.
Logarithmic Mean Calculator Calculate the logarithmic mean of two positive numbers.
Logarithmic Spiral Length Calculator Compute logarithmic spiral segment length for r=a exp(bθ), retaining shrinking spirals and exact b=0 circular limit; finite output bounds exponential overflow.
Lower Incomplete Gamma Approximation Evaluate lower integral γ(a,x) and P separately with stable bounded series/continued-fraction selection; shape>0≤170, x∈[0,10000], normalized and unnormalized values distinct, failed convergence refused.
Lucky Number Checker Test membership in the lucky-number sieve that starts with positive odd integers and repeatedly removes every kth survivor; input is capped at 100,000.
Maass Form Approximation Qualified caller-reference retrieval for Maass Form Approximation. Actual independently applicable actual Maass form/group/eigenvalue/Fourier-Whittaker coefficients/normalization/domain and independently computed certified reference. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Meissel-Mertens Constant Approximation Meissel–MertensprimeharmonicconstantB1≈0.2614972128476428. Return a bounded rounded binary64 published decimal approximation, not arbitrary precision or truncation series/error bound;0–12decimal places.
Ménage Number Calculator Count reduced circular ménage arrangements with one sex fixed in circular order and opposite-sex placements avoiding adjacent spouses; n≥2, no extra rotation/sex arrangement multiplier.
Mertens Function Return M(n)=Σ(k≤n)μ(k) exactly by sieving distinct prime factors and square factors, with M(0)=0 and n capped at 1,000,000.
Mills Constant Approximation Qualified caller-reference retrieval for Mills Constant Approximation. Actual independently applicable actual Mills constant definition and independently certified minimal branch/conditional bounds; not arbitrary prime-generating constant. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Mills Prime Generating Function Calculator Qualified caller-reference retrieval for Mills Prime Generating Function Calculator. Actual independently applicable actual independently certified Mills constant interval/iteration exponent and rounding proof; float floor alone cannot guarantee prime. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Möbius Inversion of Divisor Sums Recover f(1)…f(N) from F(n)=Σ(d|n)f(d) by exact triangular divisor-sum inversion; prefixes start at one and are capped at 1,000 entries.
Möbius Transform Calculator NumbertheoreticdivisorMobiusinversiong(n)=Σd|nμ(d) f(n/d), not complex fractional-linearMobiusmap. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Mock Theta Function Approximation ExplicitRamanujanthirdordermocktheta f(q)=Σq^(n²)/Π(1+q^k)², termsn=0…N−1, not unnamed family or analytic continuation. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Modular Discriminant Delta Calculator FiniteqΠn≤N(1−q^n)^24 standard modular discriminantΔ, realqbranch, no full infinite product error certificate, underflow rejected. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Modular Inverse Find the least nonnegative inverse of a signed decimal integer modulo a decimal modulus of at least two, rejecting noncoprime pairs.
Montgomery Pair Correlation Calculator Compute finite supplied-ordinate pair-gap histogram normalized by caller mean spacing, counting unordered i<j pairs once; does not verify ordinates are zeta zeros or prove Montgomery asymptotic pair correlation.
Morley Triangle Calculator First/interior Morley equilateral triangle SIDE8Rsin(A/3) sin(B/3) sin(C/3), strictly valid sides, same units. Not externaltrisectorvariantsor full coordinates.
Motzkin Number Count Motzkin paths of length n exactly, with M(0)=M(1)=1 and a 1,000-index resource limit.
Multinomial Coefficient Count arrangements with the supplied nonnegative category counts using exact arithmetic; at most 128 categories and 2,000 total elements are admitted.
Münchhausen Digit Self-Power Checker Check whether n equals the sum of d^d over its decimal digits under an explicit 0^0 convention of zero or one, using exact integers up to 200 input characters.
Napoleon Triangle Area Calculator Compute outer/inner Napoleon triangle areas from actual noncollinear planar vertices and erected equilateral-triangle centroids; inner construction may degenerate.
Narayana Number Count Dyck paths of semilength n with k peaks as an exact Narayana number, using N(0,0)=1 and zero outside the counting range; indices are capped at 500.
Natural Density Calculator Compute density at a finite declared integer cutoff from complete binary membership; not global prefix infimum or proof that infinite natural-density limit exists.
Nephroid Area Calculator Nephroidepicycloidarea12πa²whereaisrollingradius and fixed radius2a.
Nephroid Perimeter Calculator Nephroidepicycloidperimeter24awhereaisrollingradius.
Nine Point Circle Calculator Construct nine-point circle and actual triangle side midpoints; center uses circumcenter/orthocenter and radius is half circumradius, not a generic circle from radius.
nth Fibonacci Number Return F(n) exactly as a decimal string with F(0)=0 and F(1)=1; the index is capped at 100,000 for bounded computation.
nth Lucas Number Return L(n) exactly as a decimal string with L(0)=2 and L(1)=1; the index is capped at 100,000 for bounded computation.
nth Padovan Number Return the Padovan sequence with P(0)=P(1)=P(2)=1 and P(n)=P(n−2)+P(n−3) as an exact decimal integer; the index is capped at 10,000.
nth Pell Number Return P(n) exactly with P(0)=0, P(1)=1, and P(n)=2P(n−1)+P(n−2); the index is capped at 10,000.
Nth Prime Approximation Nthprimeasymptoticn(lnn+lnlnn), n≥6, no bound or exact prime claim.
nth Tribonacci Number Return T(n) exactly with T(0)=T(1)=0, T(2)=1, and each later term the sum of the preceding three; the index is capped at 10,000.
Odd Zeta Approximation Real positive odd zetaseriessumthroughNwithmonotoneintegral tail bounds, no analytic continuation or rounded partial as exactζ. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Odious Number Checker Odious integer has odd count of1bits in unsigned binary, zero even/not odious; arithmetic uses full safe integer not32bit truncation.
Offset Log Integral Li Calculator Approximate offset logarithmic integral Li(x)=∫2^x dt/log t by caller even-panel Simpson rule; no principal-value li(x), singularity crossing or automatic quadrature error bound.
Omega Constant Approximation Omega W0(1) solutionωe^ω=1≈0.5671432904097838. Return a bounded rounded binary64 published decimal approximation, not arbitrary precision or truncation series/error bound;0–12decimal places.
One Collatz 3n+1 Step Return one full Collatz step for a positive decimal integer: n/2 when even and 3n+1 when odd, including 1→4 and an input limit of 200 characters.
Oppermann Conjecture Checker Enumerate exact primes in both open Oppermann intervals (n²,n(n+1)) and (n(n+1),(n+1)²) for supplied n≥2; local observation, not a global proof.
Overpartition Count Count partitions of n in which the first occurrence of each distinct part may be overlined, returning an exact integer and limiting n to 1,000.
Padé Approximant Calculator Solve bounded [m/n] Padé approximant from exactly m+n+1 caller Taylor coefficients, denominator constant one; singular/unresolved systems refused rather than invent a unique approximation.
Parabolic Cylinder Approximation Qualified caller-reference retrieval for Parabolic Cylinder Approximation. Actual independently applicable actual paraboliccylinderD/U/Vchoice, order/domain/branches and independently certified special function reference. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Paraboloid Volume Calculator Compute volume under an elliptical paraboloid from measured aperture semiaxes and height; one half of aperture area times height, not cone/cylinder volume.
Partial Fraction Decomposition Tool Decompose a strictly proper rational function whose monic denominator is caller supplied product of distinct simple real (x−pole) factors; residues computed by p(pole)/q′(pole), not a general factoring claim.
Partition Crank Return the partition crank: largest part if no ones occur, otherwise the number of parts greater than the number of ones minus that number; empty crank is zero.
Partition Generating Function Approximation Evaluate truncated real generating product ∏_(k=1..N)(1−q^k)^−1 with stable log accumulation, not exact partition coefficient counting or the exponent-sequence transform.
Partition Rank Return Dyson’s rank, largest part minus number of parts, for a nonincreasing positive-integer partition; the empty partition has rank zero and input is capped at 1,000 parts.
Partition Transform Calculator Compute exact coefficients through N of formal product ∏(1−x^k)^(-a_k), allowing signed caller exponent sequence; not ordinary unrestricted partition count or finite q product evaluation.
Partitions into kth Powers Count unordered partitions of a nonnegative total into positive kth powers with repetition, using exact counts, any positive safe-integer exponent, and a total limit of 2,000.
Pascal Determinant Calculator Exact leadingN×N SYMMETRICPascalmatrixAij=binomial(i+j, i), i, j=0…N−1 has determinant1viaunittriangularLL^T,1≤N≤64; not arbitraryPascal-like matrix. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Pascal Triangle Row Return the zero-indexed nth row of Pascal’s triangle as exact decimal binomial coefficients; n is capped at 1,000 to bound the output.
Perfect Digital Invariant Checker Perfect digital invariant with independent fixed power(not automatically decimaldigitlengthasArmstrong), exactBigIntdigitpowers≤128. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Petersson Inner Product Approximation ActualPeterssonquadratureΣfconj(g) y^(k−2) dxdyontheindependentlyqualifiedfundamentaldomainnodes/Euclidean area weights, complex tail error not certified; not unweightedinnerproductor claimed modularity. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Pi from the Leibniz Series Approximate π with the requested terms of 4Σ(−1)^k/(2k+1), grouping adjacent terms to reduce cancellation; computation is capped at 100,000 terms.
Pi from the Wallis Product Approximate π by 2∏(k=1…m)4k²/(4k²−1), using the empty product value two when m=0 and a 1,000,000-term resource limit.
Pi from Viète’s Product Approximate π by 2∏(j=1…m)2/a_j with a₁=√2 and a_(j+1)=√(2+a_j), including the empty product two and a 10,000-term resource limit.
Pick Theorem Area Calculator PicklatticeareaI+B/2−1, actual independently verified simple hole free integer lattice polygon counts, integer area or half integer; counts not proof valid polygon.
Pillai Prime Checker Find exact Pillai prime witnesses k!≡−1 mod p with k not dividing p−1; exhaustive bounded k<p, no inferred classification from one arbitrary k.
Plane Partition Count Count plane partitions of n exactly from MacMahon’s generating product ∏(1−q^k)^(−k), including the empty partition; n is capped at 200.
Plastic Number Value Calculator Plastic number real positive rootx³−x−1,ρ≈1.324717957244746. Return a bounded rounded binary64 published decimal approximation, not arbitrary precision or truncation series/error bound;0–12decimal places.
Polignac Conjecture Checker Bounded off line reference: Polignac open conjecture statement/scope, no decidability claim. Topics: statement, scope. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Polygamma Approximation Approximate ψ^(m)(x) for positive x and integer m=1..6 via finite absolutely convergent reciprocal-power series, with integral-test absolute tail upper bound; finite truncation not hidden exact value.
Polynomial GCD Calculator Exact polynomial GCD over rational coefficients from bounded integer input, normalized monic; zero result is ["0"], rational coefficients are exact decimal numerator/denominator strings.
Polynomial LCM Calculator Exact polynomial LCM over rational coefficients from bounded integer input, normalized monic; zero result is ["0"], rational coefficients are exact decimal numerator/denominator strings.
Porter Constant Approximation Bounded off line reference: Porter constant bounded definition/value inventory. Topics: definition, approximate-value. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Powerful Number Checker Check whether every prime divisor of n occurs with exponent at least two, treating one as powerful; trial division is limited to n≤10¹².
Practical Number Checker Check whether every integer from one through n is a sum of distinct divisors of n using the Stewart–Sierpiński prime-factor criterion; trial division is limited to n≤10¹².
Prime Counting by the Sieve of Eratosthenes Count primes at most n exactly with the sieve of Eratosthenes, excluding zero and one and limiting n to 1,000,000.
Prime Counting Pi Approximation Crudeπ(x)≈x/lnx, declaredx≥3, not guaranteed count or probability.
Prime Density Approximation PNT local prime density heuristic1/lnx, declaredx≥3, not guaranteed count or probability.
Prime Factor Count with Multiplicity Return Ω(n), the total number of prime factors counting their exponents, with Ω(1)=0 and trial division limited to n≤10¹².
Prime Gap Calculator Exact validated consecutive prime gap not running record maximum sieve. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Prime Gap Record Calculator Find strict successive prime-gap records with both endpoints ≤caller sieve cutoff; equal ties are not new records and no outside-cutoff maximality claim.
Prime Number Theorem Approximation Finite observed PNT convergence ratioπ(x) lnx/x, NOT proof asymptotic limit, declaredx≥3, not guaranteed count or probability.
Prime Triplet Checker Check supplied prime-triplet patterns (p,p+2,p+6) or (p,p+4,p+6), plus exceptional (3,5,7); not any three primes or a twin-pair alias.
Prime Zeta Approximation Real prime zeta sum over actual primes≤cutoff, s>1, conservativeALL-integer integral tail upper bound; not sum overall integersζ. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Proth Number Checker Check whether a decimal integer is k·2^m+1 with m≥1 and positive odd k<2^m, without requiring primality; input is capped at 200 characters.
q-Binomial Coefficient Polynomial Return exact coefficients of the Gaussian binomial [n choose k]_q in ascending degree order, with zero for k>n and indices capped at 64.
q-Factorial Polynomial Return exact coefficients of [n]_q!=∏(j=1…n)(1+q+…+q^(j−1)) in ascending degree order, including the empty product one and a 64-index resource limit.
Quartic Discriminant Calculator Actual quartic a x⁴+b x³+c x²+d x+e(a≠0), Sylvester resultant(f, f′)/aleadingcoefficient(sign positive for degree4). Finite binary64determinant, no arbitrary precision or root safety.
Ramanujan Prime Checker Qualified caller-reference retrieval for Ramanujan Prime Checker. Actual independently applicable actual prime/certified nthRamanujanthreshold, allx≥R_n countπ(x)−π(x/2)≥n certificate; finite-window test insufficient. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Ramanujan Sum Return the exact integer c_q(k), the sum of exp(2πiak/q) over residues a coprime to q, using the Möbius–totient formula and q≤10¹².
Ramanujan Tau Approximation Exact Ramanujanτ(n) coefficientq^nofqΠ(1−q^k)^24via finite integer polynomial, n≤64. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Rankin-Selberg Convolution Approximation DeclaredrealRankin–Selberg finite coefficient convolutionζ(2s)Σa_nconj(b_n) n^−s, normalized level ONEH eckeeigenformsunderindependentcoefficient/weight shift convention; externally verifiedζfactor, not fullLvalue/tail certification. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Recamán Sequence Term Return a(0)=0 and a(n)=a(n−1)−n when that value is positive and unused, otherwise a(n−1)+n; the index is capped at 100,000.
Reciprocal Fibonacci Constant Approximation Compute reciprocal Fibonacci partial sum from F1=F2=1, retaining both unit terms; finite truncation only, not nth Fibonacci or a static reciprocal-Fibonacci constant.
Regular N-Gon Perimeter Calculator Regularn-gon perimeter n×s, integern≥3, not circumradius or area.
Rencontres Number Count permutations of n labeled elements with exactly k fixed points using C(n,k)·D(n−k), returning zero for k>n; indices are capped at 1,000.
Resultant of Two Quadratics Calculator Degree-two Sylvester resultant actual descending coefficients, sign conventionRes(f, g), not discriminant or fully solve common roots.
Riemann Zeta Even Approximation Real positive even zetaseriessumthroughNwithmonotoneintegral tail bounds, no analytic continuation or rounded partial as exactζ. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Rogers-Ramanujan Continued Fraction Approximation Rogers–Ramanujancontinuedfractionq^(1/5)/(1+q/(1+q²/(…))), finite explicit depth, no certified infinite convergence error. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Rose Area Calculator Compute distinct rose petals for r=a cos(kθ), integer k; odd k repeats the curve over 2π, so repeated traces are not counted as extra area.
Rosser Rule Calculator Qualified caller-reference retrieval for Rosser Rule Calculator. Actual independently applicable actual certified Gram blocks/zero counts and namedRosserrule assumptions/exceptions; no finite pattern global proof. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Rough Number Checker Check whether every prime factor of n is at least an inclusive bound B, treating one as B-rough; trial division is limited to n≤10¹².
Rough Number Count Approximation ExactfiniteBrough(all prime factors≥B) count1≤m≤N, included1by empty factor convention, N≤1e5. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Safe Prime Checker Check whether p is prime and (p−1)/2 is prime, with p capped at 10¹² for bounded trial division.
Schinzel Hypothesis H Checker Bounded off line reference: SchinzelH open conjecture statement/scope, no decidability claim. Topics: statement, scope. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Schnirelmann Basis Calculator Exactfiniteh-fold sum set of actual basis with repetition and prefix coverage, no infiniteSchnirelmannadditivebasisdensity proof. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Schnirelmann Density Calculator Compute minimum observed prefix density over explicit membership of integers 1..N; finite-prefix minimum is an upper bound on global Schnirelmann infimum, not certified infinite-set density.
Selberg Class Approximation Qualified caller-reference retrieval for Selberg Class Approximation. Actual independently applicable actual Selberg class function/axioms/gamma factors/Dirichlet normalization and independently certified evaluation; not class membership from few coefficients. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Selberg Zeta Approximation Qualified caller-reference retrieval for Selberg Zeta Approximation. Actual independently applicable actual hyperbolic surface/group/primitive geodesic lengths/multiplicities/domain and independently certified Selberg zeta product reference. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Semiprime Count Approximation Exactfinitecountsemiprimespq≤Nincludingp=q, not invalid asymptotic at smallN. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Sexy Prime Pair Checker Check whether two nonnegative integers are prime and differ by six, accepting either order and limiting each candidate to 10¹² for trial division.
Signed Stirling Number of the First Kind Return s(n,k), the cycle-count Stirling number multiplied by (−1)^(n−k), exactly as a decimal integer, with s(0,0)=1 and indices capped at 300.
Skewes Number Approximation Bounded off line reference: historicalSkewesbound symbolic reference, not floating overflow as number. Topics: scope, historical-rh-bound. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Smooth Number Checker Check whether every prime factor of n is at most an inclusive bound B, treating one as B-smooth; trial division is limited to n≤10¹².
Smooth Number Count Approximation ExactfiniteBsmooth(all prime factors≤B) count1≤m≤N, included1by empty factor convention, N≤1e5. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Sociable Chain Length 3 Checker Check exact three-cycle in proper-divisor sum iteration; fixed perfect numbers and amicable two-cycles are not mislabelled length-three sociable cycles; out-of-domain intermediate refused.
Solid Partition Count Calculator Qualified caller-reference retrieval for Solid Partition Count Calculator. Actual independently applicable actual finite solid partition size/formal array definition and independently certified combinatorial count/method; does not confuse planepartitionsMacMahonformula. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Sophie Germain Prime Checker Check whether both p and 2p+1 are prime by deterministic trial division, with p capped at 10¹² for bounded work.
Sphenic Number Checker Check whether n is the product of exactly three distinct primes, each with exponent one; trial division is limited to n≤10¹².
Star Polygon Area Calculator Regulars tar{n/k}ALGEBRAIC signed shoelace area with winding multiplicity nR²sin(2πk/n)/2, not union filled area; primitive gcd(n, k)=1and1≤k<n/2, n≥5.
Stern–Brocot Neighbor Mediant Insert the reduced mediant between ordered nonnegative Stern–Brocot neighbors satisfying b·c−a·d=1, including the boundary pair 0/1 and 1/0.
Stirling Number of the Second Kind Count partitions of n labeled elements into k nonempty unlabeled blocks exactly, returning zero when k exceeds n; indices are capped at 300.
Stirling Transform Calculator Compute exact second-kind Stirling transform b_n=Σ T(n,k)a_k, preserving integer output as strings and index-zero convention; bounded to n≤64, not a named-number lookup.
Strive Number Checker Qualified caller-reference retrieval for Strive Number Checker. Actual independently applicable actual intended definition/sourceofambiguousStrivenumber term and independently certified property reference; not silently correct to another number family. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Struve Function Approximation Evaluate Struve H_ν real power series for integer order 0..5 and x∈[0,20]; fixed family/order convention, bounded convergence, no ambiguous unnamed special function.
Subfactorial Calculator Exactderangement!nBigIntrecurrence,(n−1)(!(n−1)+!(n−2)),0≤n≤1000. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Sum of Squared Divisors Return σ₂(n), the exact sum of squares of all positive divisors, using prime-power products and trial division limited to n≤10¹².
Superfactorion Checker Exact decimal digit superfactorialS(d)=Πk! sum checker, empty product at0=1, input≤ 128 digits. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Sylvester Resultant Calculator Compute exact integer resultant by fraction-free Sylvester determinant for constant-first integer polynomials of degree ≤12; zero polynomial returns zero resultant. Nonzero constants have no common roots.
Sylvester Sequence Term Return the zero-indexed Sylvester term a(0)=2 and a(n)=a(n−1)(a(n−1)−1)+1 exactly, with an index limit of 15 to bound doubly exponential output growth.
Syracuse Function Calculator Exact acceleratedoddCollatz/Syracuse(3n+1)/2^v2, not unaccelerated Collatz single-stepor global termination. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Torus Surface Area Calculator Ring torus area4π²Rr, actualR>r>0, no spindle/horn overlap or volume.
Tractrix Length Calculator Compute finite tractrix segment x=a(t−tanh t), y=a sech t over nonnegative t from a log-cosh primitive; whole asymptotic tractrix has infinite length.
Trailing Zeros in n Factorial Count decimal trailing zeros of n! exactly by counting factors of five, accepting nonnegative decimal integers up to 200 characters.
Trimorphic Number Checker Check whether the decimal digits of a nonnegative integer are the final digits of its cube, using exact arithmetic and a 200-character input limit.
Turing Method Calculator Qualified caller-reference retrieval for Turing Method Calculator. Actual independently applicable actual Turing method contour/integral bounds/endpoint zero evidence and certified completeness certificate; observed zero list alone insufficient. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Twin Prime Conjecture Checker Bounded off line reference: Twin-prime open conjecture statement/scope, no decidability claim. Topics: statement, scope. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Twin Prime Constant Approximation Compute finite twin-prime constant Euler product over odd primes ≤cutoff using log1p; omitted factors mean finite value is not exact constant or proof of twin-prime infinitude.
Twin Prime Counting Function Count twin-prime pairs whose smaller prime is at most n, testing the upper member even when it exceeds n; the sieve limit is capped at 1,000,000.
Twin Prime Pair Checker Check whether two nonnegative integers are prime and differ by two, accepting either order and limiting each candidate to 10¹² for trial division.
Ulam Spiral Diagonal Corners and Sum Return the diagonal corner values of square spiral layer r and the total of both diagonals through that layer, counting the central one once and using exact arithmetic.
Unsigned Lah Number Count partitions of n labeled elements into k nonempty linearly ordered blocks using unsigned Lah numbers, including L(0,0)=1; indices are capped at 500.
Untouchable Number Checker Exact bounded untouchable integer0–1000underpreimagem≥1 convention;0touchedbym1,1bym2. Forn≥2 primepreimages cannot occur; everycompositemhaslargestproperdivisor≥√m, so anypreimagesatisfiesm≤n². Sieve complete bound, zero witness means absent.
Upper Incomplete Gamma Approximation Evaluate upper tail Γ(a,x) and Q separately with stable bounded series/continued-fraction selection; shape>0≤170, x∈[0,10000], normalized and unnormalized values distinct, failed convergence refused.
Vampire Number Checker Find all unordered equal-digit factor fangs whose combined decimal multiset matches candidate; both trailing-zero fangs excluded, 1–6 digit bounded search and odd lengths false.
Vandermonde Determinant 3 Calculator Compute oriented determinant of rows [1,x,x²] from three nodes without requiring caller prebuilt matrix; repeated nodes yield exact singular result.
Viswanath Constant Approximation Bounded off line reference: Viswanath constant bounded definition/value inventory. Topics: definition, approximate-value. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Von Mangoldt Lambda Function Return Λ(n)=ln(p) when n is a positive power of a prime p, and zero otherwise, with trial division limited to n≤10¹².
Wall-Sun-Sun Prime Checker Check Fibonacci-Wieferich/Wall–Sun–Sun congruence F_(p−(5/p)) mod p² for actual primes other than 2 and 5 using exact modular doubling; excluded/nonprime inputs explicitly not applicable, no existence claim.
Waring G(2) Calculator Bounded off line reference: Waring asymptotic power representation result/bounds. Topics: result, definition. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Waring G(3) Calculator Bounded off line reference: Waring asymptotic power representation result/bounds. Topics: result, definition. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Waring G(4) Calculator Bounded off line reference: Waring asymptotic power representation result/bounds. Topics: result, definition. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Waring G(5) Calculator Bounded off line reference: Waring asymptotic power representation result/bounds. Topics: result, definition. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Waring G(6) Calculator Bounded off line reference: Waring asymptotic power representation result/bounds. Topics: result, definition. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Waring Problem G(k) Calculator Bounded off line reference: WaringG(k) definition and premise correction. Topics: definition, scope. Not live/exhaustive reference, a proof of an open assertion, diagnosis or safe-use approval.
Weak Goldbach Conjecture Checker Find first nondecreasing prime triple summing to caller odd n>5 using actual sieve and bounded pair search; witnesses may repeat primes, distinct from existing strong two-prime Goldbach partition count.
Weierstrass Elliptic P Approximation Finite symmetricsquareWeierstrassplatticesum/canonical product, actualTWO full periods not half periods, cutoff≤10, poles rejected, no certified full infinite elliptic function. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Weierstrass Sigma Approximation Finite symmetricsquareWeierstrasssigmalatticesum/canonical product, actualTWO full periods not half periods, cutoff≤10, zeros retained; lattice-independent real periods required, no certified full infinite elliptic function. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Weierstrass Zeta Approximation Finite symmetricsquareWeierstrasszetalatticesum/canonical product, actualTWO full periods not half periods, cutoff≤10, poles rejected, no certified full infinite elliptic function. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Weird Number Checker Check exact weirdness as abundance with no subset of distinct proper divisors summing to n; bounded 0/1 knapsack, not abundance alone or an exhaustive integer classification beyond caller n.
Whittaker Function Approximation Qualified caller-reference retrieval for Whittaker Function Approximation. Actual independently applicable actual WhittakerM/Wchoice, complex parameters/branches/precision and independently certified special function reference. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.
Wieferich Pair Checker MutualWieferichprimes modulara^(b−1)≡1modb²andb^(a−1)≡1moda², exactBigIntnot floating powers. Explicit bounded resource/branch and actual observations or independently verified parameters; no unproved infinite assertion certified.
Wieferich Prime Checker Check whether p is prime and 2^(p−1)≡1 (mod p²) with exact modular arithmetic; primality testing is limited to p≤10¹².
Wilson Prime Checker Check whether p is prime and (p−1)!+1 is divisible by p², using modular products with a 100,000-factor resource limit.
Woodall Number Checker Check whether a nonnegative decimal integer has the form k·2^k−1 for k≥1, without requiring primality; input is capped at 200 characters.
Wright Prime Generating Function Calculator Qualified caller-reference retrieval for Wright Prime Generating Function Calculator. Actual independently applicable actual independently certified Wright tower constant interval/height and prime certificates; do not overflow tower or claim ordinary float prime generator. Does not compute an omitted physical/predictive/design model or supply authoritative standards, coefficients, norms. Examples are illustrative. Evidence truth/applicability externally verified; no diagnosis, safe-use approval, code/legal compliance or guaranteed forecast.