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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.
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
LINKS
K. Matthews, Thue's theorem and the diophantine equation x^2-D*y^2=+-N, Math. Comp. 71 (239) (2002) 1281-1286.
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-François Alcover, Oct 31 2016 *)
PROG
(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( 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
(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
CROSSREFS
Primes: A038873.
Complement of A232531. - Thomas Ordowski and Altug Alkan, Feb 09 2017
Sequence in context: A190244 A182636 A116724 * A141401 A132604 A013153
KEYWORD
nonn
AUTHOR
EXTENSIONS
Better description from Sharon Sela (sharonsela(AT)hotmail.com), Mar 10 2002
STATUS
approved

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Last modified April 15 12:32 EDT 2024. Contains 371687 sequences. (Running on oeis4.)