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

1,2

References

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

Links

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

Example

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

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 *)

Prog

(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; ) ) ) ) }

Crossrefs

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

Sequence in context: A372022 A219210 A218863 * A308645 A086889 A368162

Adjacent sequences: A082391 A082392 A082393 * A082395 A082396 A082397

Keyword

easy,nonn

Author

Cino Hilliard, Apr 14 2003

Extensions

More terms from Robert G. Wilson v, Apr 15 2003; recomputed Sep 03 2004

Status

approved