## (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   $x \; + \; 2 \, y \; \leq \; 50,000$

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

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

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

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