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A035251 Positive numbers of the form x^2 - 2y^2 with integers x, y. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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.

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

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)

EXAMPLE

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

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-Fran├žois Alcover, Oct 31 2016 *)

PROG

(PARI) direuler(p=2, 201, 1/(1-(kronecker(2, p)*(X-X^2))-X))

(PARI) {a(n)= local(m, c); if(n<1, 0, c=0; m=0; while( c<n, m++; if( sum(i=0, sqrtint(m\2), issquare(m+2*i^2)), c++)); m)} /* Michael Somos, Aug 17 2006 */

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

CROSSREFS

Cf. A035185, A042965, A001481, A000047.

Primes: A038873.

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

Sequence in context: A190244 A182636 A116724 * A141401 A132604 A013153

Adjacent sequences:  A035248 A035249 A035250 * A035252 A035253 A035254

KEYWORD

nonn,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified March 28 11:45 EDT 2017. Contains 284186 sequences.