## Equation : x^2 ± k*y^2 are square numbers, for k = 45 and 54

(1)    k = 45

$x^2 \; + \; 45 \, y^2 \; = \; a^2$
$x^2 \; - \; 45 \, y^2 \; = \; b^2$

First few primitive solutions :

$x = 41$
$y = 4$
$a = 49$
$b = 31$

$x = 3344161$
$y = 498232$
$a = 4728001$
$b = 113279$

$x = 249850594047271558364480641$
$y = 1784743287607200697430416$
$a = 250137278774864229623059201$
$b = 249563579992463717493803519$

(2)    k = 54

$x^2 \; + \; 54 \, y^2 \; = \; a^2$
$x^2 \; - \; 54 \, y^2 \; = \; b^2$

First few primitive solutions :

$x = 15$
$y = 2$
$a = 21$
$b = 3$

$x = 97281$
$y = 3780$
$a = 101169$
$b = 93231$

$x = 90154917551705707521$
$y = 6936775215861650040$
$a = 103567982890978646721$
$b = 74360548931564928321$

$x = 72814682986970084503637686819491352462727332652628889611665924101014593931284481$
$y = 9632635853193180167037014877403032847654696759341653075139264276263093523440880$
$a = 101550541241418190642331852314379726215948950703655029403565471133440448325569281$
$b = 17071721956525784003784006338821409390719049180917832503465104311651596077159681$

$x = 3603$
$y = 140$
$a = 3747$
$b = 3453$

$x = 169642382788881$
$y = 13052766376440$
$a = 194881431600081$
$b = 139922491737681$

$x = 912848563609601148866619278614228350150304047956428089921$
$y = 120760503811233216788964073372612587683082018689343652080$
$a = 1273098527704838380649758903961139897728792935927274726721$
$b = 214021351560999473737862701736069998319391182792560013121$