A057126 Numbers k such that 2 is a square mod k.

1, 2, 7, 14, 17, 23, 31, 34, 41, 46, 47, 49, 62, 71, 73, 79, 82, 89, 94, 97, 98, 103, 113, 119, 127, 137, 142, 146, 151, 158, 161, 167, 178, 191, 193, 194, 199, 206, 217, 223, 226, 233, 238, 239, 241, 254, 257, 263, 271, 274, 281, 287, 289, 302, 311, 313, 322

Offset: 1

Author

Henry Bottomley, Aug 10 2000

Keywords

• nonn

Comments

Numbers that are not multiples of 4 and for which all odd prime factors are congruent to +/- 1 mod 8. - Eric M. Schmidt, Apr 20 2013

Apparently the same as the list of numbers primitively represented by the indefinite quadratic form x^2 - 2y^2 (cf. A035251). - N. J. A. Sloane, Jun 11 2014

From Wolfdieter Lang, Jul 11 2025: (Start)

Also the negative sequence lists the numbers properly represented by the indefinite quadratic form x^2 - 2*y^2 of discriminant 4*2 = 8. For the proof see the W. Lang paper linked in A385449, Lemma 18, pp. 22-23.

The connection between the proper positive fundamental solutions (X, Y) of X^2 - 2*Y^2 = -a(n), given in A385449, and the solutions (x, y) of x^2 - 2*y^2 = a(n) is (x, y) = (2*Y - X, X - Y). If y becomes nonpositive a transformation with the matrix Mat([3,4], [2,3]) will give the positive proper fundamental solution. See the example section of A385449. See also the Nov 09 2009 comment in A035251 by Franklin T. Adams-Watters for this connection, and for the matrix eq. (38) p. 14 of the mentioned linked paper.

Therefore the previous statement on the representation of a(n) is true.(End)

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Includes the primes in A038873 and these (primes congruent to {1, 2, 7} mod 8) are the prime factors of the terms in this sequence.

Cf. A008784, A057125, A057127, A057128, A057129, A057757, A035251, A038873.

Cf. A087780 (number of solutions mod n).

Programs

Maple

with(numtheory); [seq(mroot(2,2,p),p=1..300)];

Mathematica

ok[n_] := Reduce[ Mod[2 - k^2, n] == 0, k, Integers] =!= False; Prepend[ Select[ Range[400], ok], 1] (* Jean-François Alcover, Sep 20 2012 *)

PARI

isok(n) = issquare(Mod(2,n)); \\ Michel Marcus, Feb 19 2016

Extensions

Checked by T. D. Noe, Apr 19 2007