A137351 Composite numbers n such that x^2 - n*y^2 represents -1.

10, 26, 50, 58, 65, 74, 82, 85, 106, 122, 125, 130, 145, 170, 185, 202, 218, 226, 250, 265, 274, 290, 298, 314, 325, 338, 346, 362, 365, 370, 394, 425, 442, 445, 458, 481, 485, 493, 530, 533, 538, 554, 565, 586, 610, 626, 629, 634, 685, 697, 698, 730, 746

Offset

1,1

Comments

Number of terms less than or equal to 10^k for k=0 .. : 0, 1, 8, 71, 712, 6702, 63485, 602870, ... . - Robert G. Wilson v, Jul 20 2008

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..63485

J. P. Robertson and K. R. Matthews, A continued fraction approach to a result of Feit, Amer. Math. Monthly, 115 (No. 4, 2008), 346-349.

Eric Weisstein's World of Mathematics, Pell Equation.

Example

3^2 - 10*1^2 = -1, so 10 is a member.
4005^2 - 106*389^2 = -1, so 106 is a member.

Mathematica

lst = {}; Do[ If[ !PrimeQ@ n && FindInstance[x^2 - n*y^2 == -1, {x, y}, Integers] != {}, AppendTo[lst, n]], {n, 2, 1000}]

Prog

(Python)
from itertools import count, islice
from sympy.solvers.diophantine.diophantine import diop_DN
from sympy import isprime
def A137351_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: not isprime(n) and len(diop_DN(n, -1)), count(max(startvalue, 4)))
A137351_list = list(islice(A137351_gen(), 52)) # Chai Wah Wu, Dec 22 2025

Crossrefs

For the primes with this property see A002313. A134406 is a subset.

Sequence in context: A044452 A299409 A198017 * A134406 A099978 A242719

Adjacent sequences: A137348 A137349 A137350 * A137352 A137353 A137354

Keyword

nonn

Author

N. J. A. Sloane, Apr 08 2008

Extensions

More terms from Robert G. Wilson v, Jul 20 2008

Status

approved