## Equation : x^2 ± 55*y^2 are square numbers

$x^2 \; + \; 55 \, y^2 \; = \; a^2$
$x^2 \; - \; 55 \, y^2 \; = \; b^2$

First few primitive solutions :

$17561^2 \; + \; 55\times 2340^2 \; = \; 24689^2$
$17561^2 \; - \; 55\times 2340^2 \; = \; 2689^2$

$x = 17561$
$y = 2340$
$a = 24689$
$b = 2689$

$x = 185799744204015841$
$y = 5456188901971080$
$a = 190154923153851841$
$b = 181339998601820159$

$x = 1194417980020202046744273780679344972893919507559830891813388035672961$
$y = 69914247434361269754367535147497874641977601932092353381986483686640$
$a = 1302103844047301705913626013781245581790807745156919124642161961096961$
$b = 1076008457823784835540944857292039956932975613164507454606921694248961$

other values for   $k$   and   $y$   such that

$17561^2 \; + \; k \, y^2 \; = \; 24689^2$
$17561^2 \; - \; k \, y^2 \; = \; 2689^2$

$gcd( \,17561, 1170 \,) = 1$

$17561^2 \; + \; 220\times 1170^2 \; = \; 24689^2$
$17561^2 \; - \; 220\times 1170^2 \; = \; 2689^2$

$gcd( \,17561, 780 \,) = 1$

$17561^2 \; + \; 495\times 780^2 \; = \; 24689^2$
$17561^2 \; - \; 495\times 780^2 \; = \; 2689^2$

$gcd( \,17561, 585 \,) = 1$

$17561^2 \; + \; 880\times 585^2 \; = \; 24689^2$
$17561^2 \; - \; 880\times 585^2 \; = \; 2689^2$

$gcd( \,17561,468 \,) = 1$

$17561^2 \; + \; 1375\times 468^2 \; = \; 24689^2$
$17561^2 \; - \; 1375\times 468^2 \; = \; 2689^2$

$gcd( \,17561,390 \,) = 1$

$17561^2 \; + \; 1980\times 390^2 \; = \; 24689^2$
$17561^2 \; - \; 1980\times 390^2 \; = \; 2689^2$

$gcd( \,17561,260 \,) = 1$

$17561^2 \; + \; 4455\times 260^2 \; = \; 24689^2$
$17561^2 \; - \; 4455\times 260^2 \; = \; 2689^2$

$gcd( \,17561,234 \,) = 1$

$17561^2 \; + \; 5500\times 234^2 \; = \; 24689^2$
$17561^2 \; - \; 5500\times 234^2 \; = \; 2689^2$

$gcd( \,17561,195 \,) = 1$

$17561^2 \; + \; 7920\times 195^2 \; = \; 24689^2$
$17561^2 \; - \; 7920\times 195^2 \; = \; 2689^2$

$gcd( \,17561,180 \,) = 1$

$17561^2 \; + \; 9295\times 180^2 \; = \; 24689^2$
$17561^2 \; - \; 9295\times 180^2 \; = \; 2689^2$

$gcd( \,17561,156 \,) = 1$

$17561^2 \; + \; 12375\times 156^2 \; = \; 24689^2$
$17561^2 \; - \; 12375\times 156^2 \; = \; 2689^2$

$gcd( \,17561,130 \,) = 1$

$17561^2 \; + \; 17820\times 130^2 \; = \; 24689^2$
$17561^2 \; - \; 17820\times 130^2 \; = \; 2689^2$

$gcd( \,17561,117 \,) = 1$

$17561^2 \; + \; 22000\times 117^2 \; = \; 24689^2$
$17561^2 \; - \; 22000\times 117^2 \; = \; 2689^2$

$gcd( \,17561,90 \,) = 1$

$17561^2 \; + \; 37180\times 90^2 \; = \; 24689^2$
$17561^2 \; - \; 37180\times 90^2 \; = \; 2689^2$

$gcd( \,17561,78 \,) = 1$

$17561^2 \; + \; 49500\times 78^2 \; = \; 24689^2$
$17561^2 \; - \; 49500\times 78^2 \; = \; 2689^2$

$gcd( \,17561,65 \,) = 1$

$17561^2 \; + \; 71280\times 65^2 \; = \; 24689^2$
$17561^2 \; - \; 71280\times 65^2 \; = \; 2689^2$

$gcd( \,17561,60 \,) = 1$

$17561^2 \; + \; 83655\times 60^2 \; = \; 24689^2$
$17561^2 \; - \; 83655\times 60^2 \; = \; 2689^2$

$gcd( \,17561,52 \,) = 1$

$17561^2 \; + \; 111375\times 52^2 \; = \; 24689^2$
$17561^2 \; - \; 111375\times 52^2 \; = \; 2689^2$

$gcd( \,17561,45 \,) = 1$

$17561^2 \; + \; 148720\times 45^2 \; = \; 24689^2$
$17561^2 \; - \; 148720\times 45^2 \; = \; 2689^2$

$gcd( \,17561,39 \,) = 1$

$17561^2 \; + \; 198000\times 39^2 \; = \; 24689^2$
$17561^2 \; - \; 198000\times 39^2 \; = \; 2689^2$

$gcd( \,17561,36 \,) = 1$

$17561^2 \; + \; 232375\times 36^2 \; = \; 24689^2$
$17561^2 \; - \; 232375\times 36^2 \; = \; 2689^2$

$gcd( \,17561,30 \,) = 1$

$17561^2 \; + \; 334620\times 30^2 \; = \; 24689^2$
$17561^2 \; - \; 334620\times 30^2 \; = \; 2689^2$

$gcd( \,17561,26 \,) = 1$

$17561^2 \; + \; 445500\times 26^2 \; = \; 24689^2$
$17561^2 \; - \; 445500\times 26^2 \; = \; 2689^2$

$gcd( \,17561,20 \,) = 1$

$17561^2 \; + \; 752895\times 20^2 \; = \; 24689^2$
$17561^2 \; - \; 752895\times 20^2 \; = \; 2689^2$

$gcd( \,17561,18 \,) = 1$

$17561^2 \; + \; 929500\times 18^2 \; = \; 24689^2$
$17561^2 \; - \; 929500\times 18^2 \; = \; 2689^2$

$gcd( \,17561,15 \,) = 1$

$17561^2 \; + \; 1338480\times 15^2 \; = \; 24689^2$
$17561^2 \; - \; 1338480\times 15^2 \; = \; 2689^2$

$gcd( \,17561,13 \,) = 1$

$17561^2 \; + \; 1782000\times 13^2 \; = \; 24689^2$
$17561^2 \; - \; 1782000\times 13^2 \; = \; 2689^2$

$gcd( \,17561,12 \,) = 1$

$17561^2 \; + \; 2091375\times 12^2 \; = \; 24689^2$
$17561^2 \; - \; 2091375\times 12^2 \; = \; 2689^2$

$gcd( \,17561,10 \,) = 1$

$17561^2 \; + \; 3011580\times 10^2 \; = \; 24689^2$
$17561^2 \; - \; 3011580\times 10^2 \; = \; 2689^2$

$gcd( \,17561,9 \,) = 1$

$17561^2 \; + \; 3718000\times 9^2 \; = \; 24689^2$
$17561^2 \; - \; 3718000\times 9^2 \; = \; 2689^2$

$gcd( \,17561,6 \,) = 1$

$17561^2 \; + \; 8365500\times 6^2 \; = \; 24689^2$
$17561^2 \; - \; 8365500\times 6^2 \; = \; 2689^2$

$gcd( \,17561,5 \,) = 1$

$17561^2 \; + \; 12046320\times 5^2 \; = \; 24689^2$
$17561^2 \; - \; 12046320\times 5^2 \; = \; 2689^2$

$gcd( \,17561,4 \,) = 1$

$17561^2 \; + \; 18822375\times 4^2 \; = \; 24689^2$
$17561^2 \; - \; 18822375\times 4^2 \; = \; 2689^2$

$gcd( \,17561,3 \,) = 1$

$17561^2 \; + \; 33462000\times 3^2 \; = \; 24689^2$
$17561^2 \; - \; 33462000\times 3^2 \; = \; 2689^2$

$gcd( \,17561,2 \,) = 1$

$17561^2 \; + \; 75289500\times 2^2 \; = \; 24689^2$
$17561^2 \; - \; 75289500\times 2^2 \; = \; 2689^2$

$gcd( \,17561,1 \,) = 1$

$17561^2 \; + \; 301158000\times 1^2 \; = \; 24689^2$
$17561^2 \; - \; 301158000\times 1^2 \; = \; 2689^2$