A038873 Primes p such that 2 is a square mod p; or, primes congruent to {1, 2, 7} mod 8.

2, 7, 17, 23, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137, 151, 167, 191, 193, 199, 223, 233, 239, 241, 257, 263, 271, 281, 311, 313, 337, 353, 359, 367, 383, 401, 409, 431, 433, 439, 449, 457, 463, 479, 487, 503, 521, 569, 577, 593, 599, 601, 607, 617

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Same as A001132 except for initial term.

Primes p such that x^2 = 2 has a solution mod p.

The primes of the form x^2 + 2xy - y^2 coincide with this sequence. These are also primes of the form u^2 - 2v^2. - Tito Piezas III, Dec 28 2008

Therefore these are composite in Z[sqrt(2)], as they can be factored as (u^2 - 2v^2)*(u^2 + 2v^2). - Alonso del Arte, Oct 03 2012

After a(1) = 2, these are the primes p such that p^4 == 1 (mod 96). - Gary Detlefs, Jan 22 2014

Also primes of the form 2v^2 - u^2. For example, 23 = 2*4^2 - 3^2. - Jerzy R Borysowicz, Oct 27 2015

Prime factors of A008865 and A028884. - Klaus Purath, Dec 07 2020

References

• W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, Theorem 5-5, p. 68.

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000
• K. S. Brown, Pythagorean graphs.
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
• Index entries for related sequences

Crossrefs

Cf. A057126, A087780, A226523, A003629 (complement).

Primes in A035251.

For primes p such that x^m == 2 mod p has a solution for m = 2,3,4,5,6,7,... see A038873, A040028, A040098, A040159, A040992, A042966, ...

Programs

Magma

[ p: p in PrimesUpTo(617) | IsSquare(R!2) where R:=ResidueClassRing(p) ]; // Klaus Brockhaus, Dec 02 2008

Maple

seq(`if`(member(ithprime(n) mod 8, {1,2,7}),ithprime(n),NULL),n=1..113); # Nathaniel Johnston, Jun 26 2011

Mathematica

fQ[n_] := MemberQ[{1, 2, 7}, Mod[n, 8]]; Select[ Prime[Range[114]], fQ] (* Robert G. Wilson v, Oct 18 2011 *)

PARI

is(n)=isprime(n) && issquare(Mod(2,n)) \\ Charles R Greathouse IV, Apr 23 2015

PARI

is(n)=abs(centerlift(Mod(n,8)))<3 && isprime(n) \\ Charles R Greathouse IV, Nov 14 2017

Formula

a(n) ~ 2n log n. - Charles R Greathouse IV, Nov 29 2016