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A082393 Let p = n-th prime of the form 4k+1, take the integer solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with the smallest y >= 1; sequence gives value of y.

%I A082393 #20 Jul 10 2020 03:52:33
%S A082393 4,180,8,1820,12,320,9100,226153980,267000,53000,6377352,20,
%T A082393 15140424455100,113296,519712,2113761020,3726964292220,190060,
%U A082393 183567298683461940,448036604040,28,386460,70255304,649641205044600
%N A082393 Let p = n-th prime of the form 4k+1, take the integer solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with the smallest y >= 1; sequence gives value of y.
%D A082393 C. Stanley Ogilvy, Tomorrow's Math, 1972, p. 119.
%H A082393 Vincenzo Librandi, <a href="/A082393/b082393.txt">Table of n, a(n) for n = 1..2000</a>
%H A082393 N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)
%e A082393 For n = 1, p = 5, x=9, y=4 since 9^2 = 5*4^2 + 1, so a(1) = 4.
%t A082393 PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cf]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; t = {}; Last /@ PellSolve /@ Select[Prime@Range@54, Mod[ #, 4] == 1 &] (* _Robert G. Wilson v_, Feb 28 2006 *)
%o A082393 (PARI) p4xp1(n,m) = { forstep(p=1,m,4, for(y=1,n, if(isprime(p), x=y*y*p+1; if(issquare(x), print1(y" "); break; ) ) ) ) }
%Y A082393 Values of x are in A081232. Cf. A082394, A081233, A081234. Equals A002349(p).
%K A082393 easy,nonn
%O A082393 1,1
%A A082393 _Cino Hilliard_, Apr 14 2003
%E A082393 More terms from _Robert G. Wilson v_, Feb 28 2006