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 A038873 Primes p such that 2 is a square mod p; or, primes congruent to {1, 2, 7} mod 8. 70
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Same as A001132 except for initial term. Primes p such that x^2 = 2 has a solution mod p. The prime values 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. Example 23=2X4^2-3^2. - Jerzy R Borysowicz, Oct 27 2015 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) 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], 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 STATUS approved

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Last modified June 6 17:20 EDT 2020. Contains 334829 sequences. (Running on oeis4.)