A035251 Positive numbers of the form x^2 - 2y^2 with integers x, y.

1, 2, 4, 7, 8, 9, 14, 16, 17, 18, 23, 25, 28, 31, 32, 34, 36, 41, 46, 47, 49, 50, 56, 62, 63, 64, 68, 71, 72, 73, 79, 81, 82, 89, 92, 94, 97, 98, 100, 103, 112, 113, 119, 121, 124, 126, 127, 128, 136, 137, 142, 144, 146, 151, 153, 158, 161, 162, 164, 167, 169, 175, 178

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

x^2 - 2y^2 has discriminant 8. - N. J. A. Sloane, May 30 2014

A positive number n is representable in the form x^2 - 2y^2 iff every prime p == 3 or 5 (mod 8) dividing n occurs to an even power.

Indices of nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m=2 (A035185). [amended by Georg Fischer, Sep 03 2020]

Also positive numbers of the form 2x^2 - y^2. If x^2 - 2y^2 = n, 2(x+y)^2 - (x+2y)^2 = n. - Franklin T. Adams-Watters, Nov 09 2009

Except 2, prime numbers in this sequence have the form p=8k+-1. According to the first comment, prime factors of the forms (8k+-3),(8k+-5) occur in x^2 - 2y^2 in even powers. If x^2 - 2y^2 is a prime number, those powers must be 0. Only factors 8k+-1 remain. Example: 137=8*17+1. - Jerzy R Borysowicz, Nov 04 2015

The product of any two terms of the sequence is a term too. A proof follows from the identity: (a^2-2b^2)(c^2-2d^2) = (2bd+ac)^2 - 2(ad+bc)^2. Example: 127*175 has form x^2-2y^2, with x=9335, y=6600. - Jerzy R Borysowicz, Nov 28 2015

Primitive terms (not a product of earlier terms that are greater than 1 in the sequence) are A055673 except 1. - Charles R Greathouse IV, Sep 10 2016

Positive numbers of the form u^2 + 2uv - v^2. - Thomas Ordowski, Feb 17 2017

For integer numbers z, a, k and z^2+a^2>0, k>=0: z^(4k) + a^4 is in A035251 because z^(4k) + a^4 = (z^(2k) + a^2)^2 - 2(a*z^k)^2. Assume 0^0 = 1. Examples: 3^4 + 1^4 = 82, 3^8+4^4=6817. - Jerzy R Borysowicz, Mar 09 2017

Numbers that are the difference between two legs of a Pythagorean right triangle. - Michael Somos, Apr 02 2017

Examples

			The (x,y) pairs, with minimum x, that solve the equation are (1,0), (2,1), (2,0), (3,1), (4,2), (3,0), (4,1), (4,0), (5,2), (6,3), (5,1), (5,0), (6,2), (7,3), (8,4), (6,1), (6,0), (7,2), (8,3), (7,1), (7,0), (10,5), (8,2), ... If the positive number is a perfect square, y=0 yields a trivial solution. - _R. J. Mathar_, Sep 10 2016
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• K. Matthews, Thue's theorem and the diophantine equation x^2-D*y^2=+-N, Math. Comp. 71 (239) (2002) 1281-1286.
• K. Matthews, Solving the diophantine equation x^2-D*y^2=N, D>0, (2016).
• Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998
• Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998 [Edited and cached copy]
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Cf. A035185, A042965, A001481, A000047.

Primes: A038873.

Complement of A232531. - Thomas Ordowski and Altug Alkan, Feb 09 2017

Programs

Maple

filter:= proc(n) local F;
  F:= select(t -> t[1] mod 8 = 3 or t[1] mod 8 = 5, ifactors(n)[2]);
  map(t -> t[2],F)::list(even);
end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 01 2015

Mathematica

Reap[For[n = 1, n < 200, n++, r = Reduce[x^2 - 2 y^2 == n, {x, y}, Integers]; If[r =!= False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

PARI

select(x -> x, direuler(p=2,201,1/(1-(kronecker(2,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

PARI

{a(n) = my(m, c); if( n<1, 0, c=0; m=0; while( cMichael Somos, Aug 17 2006 */

PARI

is(n)=#bnfisintnorm(bnfinit(z^2-2),n) \\ Ralf Stephan, Oct 14 2013

Python

from itertools import count, islice
from sympy import factorint
def A035251_gen(): # generator of terms
    return filter(lambda n:all(not((2 < p & 7 < 7) and e & 1) for p, e in factorint(n).items()),count(1))
A035251_list = list(islice(A035251_gen(),30)) # Chai Wah Wu, Jun 28 2022

Extensions

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Mar 10 2002