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.

4, 180, 8, 1820, 12, 320, 9100, 226153980, 267000, 53000, 6377352, 20, 15140424455100, 113296, 519712, 2113761020, 3726964292220, 190060, 183567298683461940, 448036604040, 28, 386460, 70255304, 649641205044600

Offset: 1

Author

Cino Hilliard, Apr 14 2003

Keywords

• easy
• nonn

Examples

			For n = 1, p = 5, x=9, y=4 since 9^2 = 5*4^2 + 1, so a(1) = 4.
		

References

• C. Stanley Ogilvy, Tomorrow's Math, 1972, p. 119.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..2000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Crossrefs

Values of x are in A081232. Cf. A082394, A081233, A081234. Equals A002349(p).

Programs

Mathematica

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 *)

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; ) ) ) ) }

Extensions

More terms from Robert G. Wilson v, Feb 28 2006