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.

Description

The twin-prime constant is represented by an Euler product over odd primes.

The twin-prime constant is represented by an Euler product over odd primes. A finite cutoff omits factors and must be reported as a finite product.^[1]

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.

When to use Twin Prime Constant Approximation

Use this tool to 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.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use Twin Prime Constant Approximation and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter primeCutoff using the field table and calculation rules, then run the tool.
Read finiteTwinPrimeEulerProduct, includedOddPrimeCount. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
primeCutoff	Prime cutoff	number	Required

Enter an integer prime cutoff from 0 to one million. Sieve actual primes, omit 2, accumulate logarithms with log1p and exponentiate. Return the product and included odd-prime count; no included primes gives product 1.

finite product = ∏odd primes p≤cutoff [1−1/(p−1)²]

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
finiteTwinPrimeEulerProduct	Finite twin prime euler product	number
includedOddPrimeCount	Included odd prime count	number

json

{
  "input": {
    "primeCutoff": 7
  },
  "output": {
    "finiteTwinPrimeEulerProduct": 0.68359375,
    "includedOddPrimeCount": 3
  }
}

Limitations and assumptions

No omitted-tail estimate, exact constant or conclusion about infinitely many twin primes is supplied.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use 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. For a complementary task, use 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.

References

Twin Prime Constant Approximation — tool.io

Similar or alternative tools

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.
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.