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

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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