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.

Description

Cramér's conjecture concerns asymptotic prime-gap growth relative to logarithmic scale.

Cramér's conjecture concerns asymptotic prime-gap growth relative to logarithmic scale. A caller's finite screening coefficient is a policy, rather than a universal coefficient-one theorem.^[1]

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.

When to use Cramér Conjecture Checker

Use this tool to 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.

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

How to use Cramér 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, callerGapToLogSquaredScreenLimit using the field table and calculation rules, then run the tool.
Read observedGapToLogSquaredRatios, aboveCallerScreenLeftPrimes. 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
callerGapToLogSquaredScreenLimit	Caller gap to log squared screen limit	number	Required

Use integer sieve cutoff 0–one million and a nonnegative caller screen limit. Return every observed adjacent-gap ratio and left primes whose ratios strictly exceed the limit. Equality does not screen.

gap ratio = (next prime−prime)/(log prime)²

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

Output field	Meaning	Value type
observedGapToLogSquaredRatios	Observed gap to log squared ratios	list of number
aboveCallerScreenLeftPrimes	Above caller screen left primes	list of number

json

{
  "input": {
    "primeCutoff": 13,
    "callerGapToLogSquaredScreenLimit": 1
  },
  "output": {
    "observedGapToLogSquaredRatios": [
      2.0813689810056077,
      1.6570708993804466,
      0.7721142018083943,
      1.0563660251615101,
      0.3478320309940516
    ],
    "aboveCallerScreenLeftPrimes": [
      2,
      3,
      7
    ]
  }
}

Limitations and assumptions

A finite exceedance neither proves nor disproves an asymptotic big-O conjecture. Small cutoffs can validly produce empty lists.

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 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. For a complementary task, 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.

References

Cramér Conjecture Checker — tool.io

Similar or alternative tools

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.
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.