A082394 Let p = n-th prime of the form 4k+3, take the solution (x,y) to the Pellian equation x^2 - p*y^2 = 1 with x and smallest y >= 1; sequence gives value of y.

1, 3, 3, 39, 5, 273, 531, 7, 69, 5967, 413, 9, 9, 22419, 93, 419775, 927, 6578829, 140634693, 5019135, 13, 313191, 650783, 1153080099, 19162705353, 15, 15, 400729, 231957, 8579, 7044978537, 8219541, 5052633, 957397, 153109862634573, 34443, 19

Offset: 1

Author

Cino Hilliard, Apr 14 2003

Keywords

• easy
• nonn

Examples

			For n=3, p = 11, x=10, y=3 since we have 10^2 = 11*3^2 + 1, so a(3) = 3.
		

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1990

Crossrefs

Values of x are in A081231. Equals A002349(p). Cf. A082393.

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]}]; Transpose[ PellSolve /@ Select[ Prime[ Range[72]], Mod[ #, 4] == 3 &]][[2]] (* Robert G. Wilson v, Sep 02 2004 *)

PARI

p4xp3(n,m) = { forstep(p=3,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, Apr 15 2003; recomputed Sep 03 2004