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.

Description

A Pillai-prime witness is a factorial index satisfying a modular condition plus a divisibility exclusion.

A Pillai-prime witness is a factorial index satisfying a modular condition plus a divisibility exclusion. Exhaustively checking bounded indices avoids treating one arbitrary failed index as a classification.^[1]

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.

When to use Pillai Prime Checker

Use this tool to 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.

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

How to use Pillai Prime 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 candidatePrime using the field table and calculation rules, then run the tool.
Read isPrimeCandidate, witnessFactorialIndices, isPillaiPrime. 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
candidatePrime	Candidate prime	number	Required

Enter an integer candidate from 2 to 10,000. Test primality; for primes accumulate factorial residues for 1≤k<p and collect every qualifying index. Composites return no witnesses and false.

witness k: k! mod p = p−1 and k does not divide p−1

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

Output field	Meaning	Value type
isPrimeCandidate	Is prime candidate	boolean
witnessFactorialIndices	Witness factorial indices	list of number
isPillaiPrime	Is pillai prime	boolean

json

{
  "input": {
    "candidatePrime": 23
  },
  "output": {
    "isPrimeCandidate": true,
    "witnessFactorialIndices": [
      14,
      18
    ],
    "isPillaiPrime": true
  }
}

Limitations and assumptions

This checks the specified bounded candidate. It makes no claim about all 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 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. 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

Pillai Prime Checker — tool.io

Similar or alternative tools

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.
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.