(x + 2y) and (x^2 + 2*y^2) are both squares

 
 
x \; + \; 2 \, y \; = \; A^2
x^2 \; + \; 2 \, y^2 \; = \; B^2

gcd(x, y) \; = \; 1
 
 
Here are all values of   (x, y)   for   x \; + \; 2 \, y \; \leq \; 50,000
 

1 + 2(12) = 5^2 …………… 1^2 + 2(12^2) = 17^2

257 + 2(136) = 23^2 ……….. 257^2 + 2(136^2) = 321^2
217 + 2(204) = 25^2 ……….. 217^2 + 2(204^2) = 361^2
233 + 2(304) = 29^2 ……….. 233^2 + 2(304^2) = 489^2
337 + 2(756) = 43^2 ……….. 337^2 + 2(756^2) = 1121^2
889 + 2(660) = 47^2 ……….. 889^2 + 2(660^2) = 1289^2

1777 + 2(516) = 53^2 ………. 1777 + 2(516^2) = 1921^2
1049 + 2(1720) = 67^2 ……… 1049 + 2(1720^2) = 2649^2
3049 + 2(1140) = 73^2 ……… 3049 + 2(1140^2) = 3449^2
6433 + 2(1296) = 95^2 ……… 6433 + 2(1296^2) = 6689^2
4937 + 2(2044) = 95^2 ……… 4937 + 2(2044^2) = 5721^2
5921 + 2(3652) = 115^2 …….. 5921 + 2(3652^2) = 7857^2
9289 + 2(3168) = 125^2 …….. 9289 + 2(3168^2) = 10313^2
8233 + 2(5544) = 139^2 …….. 8233 + 2(5544^2) = 11369^2

14249 + 2(3388) = 145^2 ……. 14249 + 2(3388^2) = 15033^2
2513 + 2(9256) = 145^2 …….. 2513 + 2(9256^2) = 13329^2
16961 + 2(2620) = 149^2 ……. 16961 + 2(2620^2) = 17361^2
13777 + 2(6396) = 163^2 ……. 13777 + 2(6396^2)  = 16481^2
4249 + 2(12840) = 173^2 ……. 4249 + 2(12840^2) = 18649^2
7441 + 2(14520) = 191^2 ……. 7441 + 2(14520^2) = 21841^2
25769 + 2(6520) = 197^2 ……. 25769 + 2(6520^2) = 27369^2
6793 + 2(18864) = 211^2 ……. 6793 + 2(18864^2) = 27529^2
36961 + 2(4632) = 215^2 ……. 36961 + 2(4632^2) = 37537^2
10057 + 2(18084) = 215^2 …… 10057 + 2(18084^2) = 27481^2

 
 

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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