a^2 + ab + b^2 = c^2 + cd + d^2; a^4 + (ab)^2 + b^4 = c^4 + (cd)^2 + d^4

$a^2 \; + \; a \, b \; + \; b^2 \; = \; c^2 \; + \; c \, d \; + \; d^2$
$2 \,(a^2 \; + \; a \, b \; + \; b^2) \; = \; 2 \,(c^2 \; + \; c \, d \; + \; d^2)$
$2 \, a^2 \; + \; 2 \, a \, b \; + \; 2 \, b^2 \; = \; 2 \, c^2 \; + \; 2 \, c \, d \; + \; 2 \, d^2$
$a^2 \; + \; b^2 \; + \; (a+b)^2 \; = \; c^2 \; + \; d^2 \; + \; (c + d)^2$

or equivalently,

$2 \, (a^2 \; + \; a \, b \; + \; b^2)^2 \; = \; 2 \, (c^2 \; + \; c \, d \; + \; d^2)^2$
that is,
$a^4 \; + \; b^4 \; + \; (a+b)^4 \; = \; c^4 \; + \; d^4 \; + \; (c+d)^4$

For example,

$6^2 \; + \; (6\times 5) \; + \; 5^2 \; = \; 9^2 \; + \; (9\times 1) \; + \; 1^2 = 91$

$6^2 \; + \; 5^2 \; + \; 11^2 \; = \; 9^2 \; + \; 1^2 \; + \; 10^2 \; = \; 182$
$6^4 \; + \; 5^4 \; + \; 11^4 \; = \; 9^4 \; + \; 1^4 \; + \; 10^4 \; = \; 16562$

$(a^2)^2 + (a^2 \, b^2) + (b^2)^2 \; = \; (c^2)^2 + (c^2 \, d^2) + (d^2)^2$
$2 (a^2)^2 + 2 (a^2 \, b^2) + 2 (b^2)^2 \; = \; 2 (c^2)^2 + 2 (c^2 \, d^2) + 2 (d^2)^2$
$(a^2)^2 + (b^2)^2 + (a^2 + b^2)^2 \; = \; (c^2)^2 + (d^2)^2 + (c^2 + d^2)^2$
$(a^2)^4 + (b^2)^4 + (a^2 + b^2)^4 \; = \; (c^2)^4 + (d^2)^4 + (c^2 + d^2)^4$

for example,

$(22^2)^2 + (17^2)^2 + (22^2 + 17^2)^2 \; = \; (26^2)^2 + (1^2)^2 + (26^2 + 1^2)^2$
$484^2 + 289^2 + 773^2 \; = \; 676^2 + 1^2 + 677^2 \; = \; 915306$
$484^4 + 289^4 + 773^4 \; = \; 676^4 + 1^4 + 677^4 \; = \; 418892536818$

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About benvitalis

math grad - Interest: Number theory
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