files / problems / brocard / compute / q3 / offset-certificate.json

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{
  "claim": "For each row, no Brown index n=p-c exists when p is prime in the stated class modulo 8. Each family is infinite by Dirichlet.",
  "nonresidue_fact": "2 is a non-square modulo primes p = 3 or 5 (mod 8).",
  "proof_identity": "If p=n+c is prime, Wilson gives n! = (-1)^c / (c-1)! (mod p).",
  "rows": [
    {
      "first_sample_prime": 3,
      "index_residue_mod_8": 1,
      "last_sample_prime": 9931,
      "offset": 2,
      "prime_residue_mod_8": 3,
      "sample_prime_count": 311,
      "target_denominator": 1,
      "target_numerator": 2
    },
    {
      "first_sample_prime": 5,
      "index_residue_mod_8": 3,
      "last_sample_prime": 9973,
      "offset": 2,
      "prime_residue_mod_8": 5,
      "sample_prime_count": 314,
      "target_denominator": 1,
      "target_numerator": 2
    },
    {
      "first_sample_prime": 11,
      "index_residue_mod_8": 0,
      "last_sample_prime": 9931,
      "offset": 3,
      "prime_residue_mod_8": 3,
      "sample_prime_count": 310,
      "target_denominator": 2,
      "target_numerator": 1
    },
    {
      "first_sample_prime": 5,
      "index_residue_mod_8": 2,
      "last_sample_prime": 9973,
      "offset": 3,
      "prime_residue_mod_8": 5,
      "sample_prime_count": 314,
      "target_denominator": 2,
      "target_numerator": 1
    }
  ],
  "sample_prime_bound": 10000,
  "schema_version": 1
}