A137351 - OEIS

%I #17 Feb 16 2025 08:33:07

%S 10,26,50,58,65,74,82,85,106,122,125,130,145,170,185,202,218,226,250,

%T 265,274,290,298,314,325,338,346,362,365,370,394,425,442,445,458,481,

%U 485,493,530,533,538,554,565,586,610,626,629,634,685,697,698,730,746

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

%C 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

%H Robert G. Wilson v, <a href="/A137351/b137351.txt">Table of n, a(n) for n = 1..63485</a>

%H J. P. Robertson and K. R. Matthews, <a href="http://www.jstor.org/stable/27642477">A continued fraction approach to a result of Feit</a>, Amer. Math. Monthly, 115 (No. 4, 2008), 346-349.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PellEquation.html">Pell Equation</a>.

%e 3^2 - 10*1^2 = -1, so 10 is a member.

%e 4005^2 - 106*389^2 = -1, so 106 is a member.

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

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

%K nonn,changed

%O 1,1

%A _N. J. A. Sloane_, Apr 08 2008

%E More terms from _Robert G. Wilson v_, Jul 20 2008