## a^n + b^n + c^n = x^n + y^n + z^n, n = 1,2

(1)

$1 \; + \; 11 \; + \; 24 \; = \; 3 \; + 8 \; + \; 25 \; = \; 6^2$
$1^2 \; + \; 11^2 \; + \; 24^2 \; = \; 3^2 \; + \; 8^2 \; + \; 25^2 \; = \; 698$

$3 \; + 11 \; + \; 22 \; = \; 6 \; + \; 7 \; + \; 23 \; = \; 6^2$
$3^2 \; + \; 11^2 \; + \; 22^2 \; = \; 6^2 \; + \; 7^2 \; + \; 23^2 \; = \; 614$

$1 \; + 17 \; + \; 18 \; = \; 2 \; + \; 13 \; + \; 21 \; = \; 6^2$
$1^2 \; + \; 17^2 \; + \; 18^2 \; = \; 2^2 \; + \; 13^2 \; + \; 21^2 \; = \; 614$

N.B.   $614$   is the smallest number with exactly 9 representations as a sum of 3 positive squares:

$614 = 1^2 + 17^2 + 18^2$
$614 = 2^2 + 9^2 + 23^2$
$614 = 2^2 + 13^2 + 21^2$
$614 = 3^2 + 11^2 + 22^2$
$614 = 6^2 + 7^2 + 23^2$
$614 = 6^2 + 17^2 + 17^2$
$614 = 7^2 + 9^2 + 22^2$
$614 = 10^2 + 15^2 + 17^2$
$614 = 11^2 + 13^2 + 18^2$

$4 \; + 14 \; + \; 31 \; = \; 7 \; + \; 10 \; + \; 32 \; = \; 7^2$
$4^2 \; + \; 14^2 \; + \; 31^2 \; = \; 7^2 + 10^2 + 32^2 \; = \; 1173$

$1 \; + 16 \; + 32 \; = \; 5 \; + \; 10 \; + \; 34 \; = \; 7^2$
$1^2 \; + \; 16^2 \; + \; 32^2 \; = \; 5^2 \; + 10^2 \; + \; 34^2 \; = \; 1281$

$2 \; + 18 \; + 29 \; = \; 8 \; + 9 \; + \; 32 \; = \; 7^2$
$2^2 \; + \; 18^2 \; + \; 29^2 \; = \; 8^2 \; + \; 9^2 \; + \; 32^2 \; = \; 1169$

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(2)

$1 \; + \; 18 \; + \; 30 \; = \; 6 + 10 + 33 \; = \; 7^2$
$1^2 \; + \; 18^2 \; + \; 30^2 \; = \; 6^2 \; + \; 10^2 \; + \; 33^2 \; = \; 1225 \; = \; 35^2$

N.B.   $1225$   is the smallest number with 3 representations as a sum of 4 positive cubes:
$1225 = 1^3+2^3+6^3+10^3$
$1225 = 3^3+7^3+7^3+8^3$
$1225 = 4^3+6^3+6^3+9^3$

Find more solutions.

Paul found:

$18 \; + \; 34 \; + \; 69 \; = \; 21 \; + 30 \; + \; 70 \; = \; 11^2$
$18^2 \; + \; 34^2 \; + \; 69^2 \; = \; 21^2 \; + \; 30^2 \; + \; 70^2 \; = \; 79^2$

$25 \; + \; 68 \; + \; 76 \; = \; 40 \; + \; 41 \; + \; 88 \; = \; 13^2$
$25^2 \; + \; 68^2 \; + \; 76^2 \; = \; 40^2 \; + \; 41^2 \; + \; 88^2 \; = \; 105^2$

math grad - Interest: Number theory
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### 2 Responses to a^n + b^n + c^n = x^n + y^n + z^n, n = 1,2

1. Paul says:

A few more Pt2

$18 + 34 + 69 = 21 + 30 + 70 = 121$
$18^2 + 34^2 + 69^2 = 21^2 + 30^2 + 70^2 = 6241$

$25 + 68 + 76 = 40 + 41 + 88 = 169$
$25^2 + 68^2 + 76^2 = 40^2 + 41^2 + 88^2 = 11025$

Paul.