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.

Description

Andrica's inequality concerns square-root gaps between consecutive primes.

Andrica's inequality concerns square-root gaps between consecutive primes. A rationalized expression avoids subtracting nearby square roots.^[1]

Check finite Andrica √p_next−√p<1 using stable gap/(√p_next+√p); actual prime sieve, no inference about all future pairs.

When to use Andrica Conjecture Checker

Use this tool to check finite Andrica √p_next−√p<1 using stable gap/(√p_next+√p); actual prime sieve, no inference about all future pairs.

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

How to use Andrica Conjecture Checker 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 observedSquareRootPrimeGaps, andricaViolationLeftPrimes, allObservedPairsSatisfyInequality. 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

Use integer sieve cutoff 0–one million. Compute every observed adjacent square-root gap and collect left primes where the value is at least 1. Return a satisfaction flag; an empty observed inventory satisfies it vacuously.

sqrt(next prime)−sqrt(prime)=gap/[sqrt(next prime)+sqrt(prime)]

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

Output field	Meaning	Value type
observedSquareRootPrimeGaps	Observed square root prime gaps	list of number
andricaViolationLeftPrimes	Andrica violation left primes	list of number
allObservedPairsSatisfyInequality	All observed pairs satisfy inequality	boolean

json

{
  "input": {
    "primeCutoff": 7
  },
  "output": {
    "observedSquareRootPrimeGaps": [
      0.3178372451957822,
      0.5040171699309124,
      0.40968333356480086
    ],
    "andricaViolationLeftPrimes": [],
    "allObservedPairsSatisfyInequality": true
  }
}

Limitations and assumptions

Finite satisfaction gives no conclusion about all larger primes.

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. A successfully returned false flag is an evaluated outcome; its meaning is explained above.

Alternative or Complementary methods

Use 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. For a complementary task, use 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.

References

Andrica Conjecture Checker — tool.io

Similar or alternative tools

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.
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.