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

Author

Steven Lu, Feb 16 2025

Keywords

• nonn

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.

Examples

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

Links

• Steven Lu, Congruence classes modulo 9240 of the terms

Programs

Mathematica

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

PARI

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