A377476 Primes p such that 1..12 are quadratic residues modulo p.

479, 1151, 1319, 1559, 2351, 2689, 2999, 3529, 3671, 3911, 4751, 4919, 5519, 5569, 5711, 6551, 6599, 7559, 7561, 7681, 8089, 8761, 8951, 9239, 9241, 9601, 9719, 9769, 10391, 10559, 10799, 12049, 12239, 12721, 12911, 13151, 13729, 14159, 14281, 14759, 14951, 15671, 15791, 16631, 16921

Offset

1,1

Comments

An odd prime p is a term if and only if the Legendre symbol Legendre(q|p) = 1 for all q = 2,3,5,7,11; i.e., each prime q <= 12 is a quadratic residue.

Prime p is a term if and only if all the following conditions are satisfied:

p == +-1 (mod 24)

p == +-1 (mod 10)

p == +-1, +-3, +-9 (mod 28)

p == +-1, +-5, +-7, +-9, +-19 (mod 44)

Prime p is a term if and only if it is congruent to any number in the attached file modulo 9240.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Steven Lu, Congruence classes modulo 9240 of the terms

Example

479 is a term of this sequence, since Legendre(b|479) = 1 for b = 1, 2, ..., 12.

Mathematica

Select[Prime /@ Range[2000], And @@ Table[KroneckerSymbol[b, #] == 1, {b, Range[12]}] &]

Prog

(PARI) isok(p)={for(i=1, 12, if(kronecker(i, p)<0, return(0))); isprime(p)} \\ Andrew Howroyd, Feb 17 2025

Crossrefs

Sequence in context: A323051 A025025 A395529 * A108256 A393084 A262222

Adjacent sequences: A377473 A377474 A377475 * A377477 A377478 A377479

Keyword

nonn

Author

Steven Lu, Feb 16 2025

Status

approved