A130226 Smallest integer x satisfying the Pell equation x^2-k*y^2=-1 for the values of k given in A031396.

0, 1, 2, 3, 18, 4, 5, 70, 6, 32, 7, 182, 99, 29718, 8, 1068, 43, 9, 378, 500, 5604, 10, 4005, 8890182, 776, 11, 682, 57, 1744, 12, 113582, 4832118, 13, 1118, 1111225770, 68, 1764132, 14, 3141, 251, 15, 1710, 23156, 71011068, 4443, 16, 6072, 82, 1407

Offset

1,3

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

K. Lakshmi and R. Someshwari, On The Negative Pell Equation y^2 = 72x^2 - 23, International Journal of Emerging Technologies in Engineering Research (IJETER), Volume 4, Issue 7, July (2016).

R. Suganya and D. Maheswari, On the Negative Pellian Equation y^2 = 110 * x^2 - 29, Journal of Mathematics and Informatics, Vol. 11 (2017), pp. 63-71.

S. Vidhyalakshmi, V. Krithika, and K. Agalya, On The Negative Pell Equation y^2 = 72x^2 - 8, International Journal of Emerging Technologies in Engineering Research (IJETER) Volume 4, Issue 2, February (2016).

Example

a(5)=18 because A031396(5)=13, and the solution to x^2-13y^2=-1 with smallest possible x has x=18.

Maple

A130226 := proc(m)
    local xm, x , i, xmo, y2;
    xm := [] ; # x^2-m*y^2=-1 (mod m) requires x in xm[]
    for x from 0 to m-1 do
        if modp(x^2, m) = modp(-1, m) then
            xm := [op(xm), x] ;
        end if;
    end do:
    for i from 0 do
        for xmo in xm do
            x := i*m+xmo ;
            y2 := (x^2+1)/m ;
            if issqr(y2) then
                return x ;
            end if;
        end do:
    end do:
end proc:
L := BFILETOLIST("b031396.txt") ;
n := 1:
for m in L do
    printf("%d %d\n", n, A130226(m)) ;
    n := n+1 ;
end do: # R. J. Mathar, Oct 19 2014

Mathematica

terms = 1000;
a031396 = Cases[Import["https://oeis.org/A031396/b031396.txt", "Table"], {_, _}][[;; terms, 2]];
sol[n_] := Solve[x > 0 && y > 0 && x^2 - n y^2 == -1, {x, y}, Integers];
a[1] = 0; a[n_] := a[n] = x /. sol[a031396[[n]]] /. C[1] -> 0 // First // Simplify // Quiet;
Table[Print[n, " ", a031396[[n]], " ", a[n]]; a[n], {n, 1, terms}] (* Jean-François Alcover, Apr 05 2020 *)

Crossrefs

Cf. A094048.

Sequence in context: A263048 A118702 A073524 * A287621 A348279 A344530

Adjacent sequences: A130223 A130224 A130225 * A130227 A130228 A130229

Keyword

nonn,changed

Author

Colin Barker, Aug 05 2007

Status

approved