A225946 Nonsquare k such that the minimal (in y) solution 0 < y < x of x^2 - k*y^2 = 1 has x-y square.

2, 3, 17, 24, 30, 40, 44, 84, 87, 99, 130, 182, 260, 288, 442, 448, 635, 650, 672, 675, 888, 894, 1211, 1299, 1368, 1605, 1616, 1722, 1748, 1955, 2034, 2499, 2541, 3150, 3287, 3782, 4224, 4400, 4920, 5073, 5619, 6723, 7242, 7310, 8487, 9228, 10200, 11055

Offset: 1

Author

Irina Gerasimova, May 21 2013

Keywords

• nonn

Comments

Numbers n such that A002350(n) - A002349(n) is a nonzero square. - Charles R Greathouse IV, Jun 06 2013

Examples

			3^2 - 2*2^2 = 1 and 3 - 2 = 1 (square), so a(1) = 2;
2^2 - 3*1^2 = 1 and 2 - 1 = 1 (square), so a(2) = 3;
33^2 - 17*8^2 = 25 and 33 - 8 = 25 (square), so a(3) = 17.
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..192

Crossrefs

Cf. A000037, A033313, A033317.

Programs

Mathematica

qQ[n_] := IntegerQ@Sqrt@n; Select[Range[500], ! qQ[#] && qQ[(x - y) /. ToRules[Expand[ Reduce[x^2 - #*y^2 == 1 && x>0 && y>0, {x,y}, Integers] /. C[1] -> 1]]] &] (* Giovanni Resta, May 25 2013 *)

PARI

is(n)=if(issquare(n),return(0));my(cf=contfrac(sqrt(n)),t,N,D);for(i=1,#cf-1,t=cf[i+1];forstep(j=i,1,-1,t=cf[j]+1/t);N=numerator(t);D=denominator(t);if(N^2-n*D^2==1,return(issquare(N-D)))); warning("Insufficient precision for "n) \\ Charles R Greathouse IV, Jun 06 2013

Extensions

a(15)-a(47) from Giovanni Resta, May 25 2013