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

Author

Colin Barker, Aug 05 2007

Keywords

• nonn

Examples

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

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).

Crossrefs

Cf. A094048.

Programs

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 *)