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

1,1

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

Formula

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

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 *)

Prog

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

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, ...

Sequence in context: A074884 A227144 A105911 * A141131 A049594 A049590

Adjacent sequences: A038870 A038871 A038872 * A038874 A038875 A038876

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved