On Fri, 2 May 2003, David Wilson wrote:

When I solved

[1] 8 x^2 - 343 y^2 = 1,

all solutions (x, y) were of the form (26041 j, 3397 k).

Shouldn't that be 3977j

An similar situation appears to be true of

22 x^2 - 29 y^2 = 1,

where all solutions (x, y) seem to be of the form (31 j, 27 k).

For which (a, b) do all solutions of

a x^2 - b y^2 = 0

have the form (r k, s j) for some fixed r, s > 1?

Maple gives parametrized solutions for the diophantine equations a*x^2-b*y^2=1. Trying a from 1 to 20 and b from 1 to 50 and taking the gcd of the first 30 solutions for x and the gcd of the first 30 solutions for y I get the following possible values of s and t: a,b = 2, 23 r,s = 78, 23 a,b = 2, 47 r,s = 732, 151 a,b = 3, 23 r,s = 36, 13 a,b = 4, 7 r,s = 4, 3 a,b = 4, 11 r,s = 5, 3 a,b = 4, 19 r,s = 85, 39 a,b = 4, 23 r,s = 12, 5 a,b = 4, 27 r,s = 13, 5 a,b = 4, 31 r,s = 760, 273 a,b = 4, 43 r,s = 1741, 531 a,b = 4, 47 r,s = 24, 7 a,b = 5, 11 r,s = 3, 2 a,b = 5, 31 r,s = 5, 2 a,b = 5, 41 r,s = 63, 22 a,b = 6, 29 r,s = 11, 5 a,b = 6, 47 r,s = 14, 5 a,b = 7, 3 r,s = 2, 3 a,b = 7, 19 r,s = 430, 261 a,b = 7, 31 r,s = 524, 249 a,b = 7, 38 r,s = 7, 3 a,b = 7, 47 r,s = 412, 159 a,b = 8, 23 r,s = 39, 23 a,b = 8, 47 r,s = 366, 151 a,b = 9, 5 r,s = 3, 4 a,b = 9, 14 r,s = 5, 4 a,b = 9, 17 r,s = 11, 8 a,b = 9, 20 r,s = 3, 2 a,b = 9, 23 r,s = 8, 5 a,b = 9, 26 r,s = 17, 10 a,b = 9, 29 r,s = 3267, 1820 a,b = 9, 41 r,s = 683, 320 a,b = 9, 47 r,s = 16, 7 a,b = 9, 50 r,s = 33, 14 a,b = 10, 31 r,s = 206, 117 a,b = 11, 2 r,s = 3, 7 a,b = 11, 7 r,s = 4, 5 a,b = 11, 19 r,s = 46, 35 a,b = 11, 35 r,s = 66, 37 a,b = 11, 50 r,s = 1179, 553 a,b = 12, 23 r,s = 18, 13 a,b = 13, 1 r,s = 5, 18 a,b = 13, 4 r,s = 5, 9 a,b = 13, 9 r,s = 5, 6 a,b = 13, 23 r,s = 4, 3 a,b = 13, 27 r,s = 49, 34 a,b = 13, 29 r,s = 3, 2 a,b = 13, 36 r,s = 5, 3 a,b = 13, 43 r,s = 4414, 2427 a,b = 14, 5 r,s = 3, 5 a,b = 15, 11 r,s = 6, 7 a,b = 16, 7 r,s = 2, 3 a,b = 16, 23 r,s = 6, 5 a,b = 16, 31 r,s = 380, 273 a,b = 16, 47 r,s = 12, 7 a,b = 17, 13 r,s = 7, 8 a,b = 17, 38 r,s = 3, 2 a,b = 18, 23 r,s = 26, 23 a,b = 18, 47 r,s = 244, 151 a,b = 19, 3 r,s = 2, 5 a,b = 19, 10 r,s = 37, 51 a,b = 19, 15 r,s = 8, 9 a,b = 19, 27 r,s = 602, 505 a,b = 19, 31 r,s = 1044070552, 817383375 a,b = 19, 50 r,s = 73, 45 ---See any patterns there? Edwin