## a^n + b^n + c^n = x^n + y^n + z^n, n = 1,3

Find a few more solutions.

the smallest integer that can be expressed …

in 2 ways :     26
in 3 ways :     41
in 4 ways :     43

Paul found:

$4 + 16 + 20 = 8 + 10 + 22 \; ....................... \; ( \,= 40 \,)$
$4^3 + 16^3 + 20^3 = 8^3 + 10^3 + 22^3 = 12160$

$1 + 17 + 23 = 5 + 11 + 25 \; ....................... \; ( \,= 41 \,)$
$1^3 + 17^3 + 23^3 = 5^3 + 11^3 + 25^3 = 17081$

$2 + 17 + 22 = 3 + 15 + 23 \; ....................... \; ( \,= 41 \,)$
$2^3 + 17^3 + 22^3 = 3^3 + 15^3 + 23^3 = 15569$

$5 + 16 + 20 = 6 + 14 + 21 \; ....................... \; ( \,= 41 \,)$
$5^3 + 16^3 + 20^3 = 6^3 + 14^3 + 21^3 = 12221$

$4 + 17 + 21 = 7 + 12 + 23 \; ....................... \; ( \,= 42 \,)$
$4^3 + 17^3 + 21^3 = 7^3 + 12^3 + 23^3 = 14238$

$4 + 18 + 20 = 9 + 10 + 23 \; ....................... \; ( \,= 42 \,)$
$4^3 + 18^3 + 20^3 = 9^3 + 10^3 + 23^3 = 13896$

$7 + 15 + 20 = 9 + 12 + 21 \; ....................... \; ( \,= 42 \,)$
$7^3 + 15^3 + 20^3 = 9^3 + 12^3 + 21^3 = 11718$

$1 + 19 + 23 = 3 + 15 + 25 \; ....................... \; ( \,= 43 \,)$
$1^3 + 19^3 + 23^3 = 3^3 + 15^3 + 25^3 = 19027$

$3 + 16 + 24 = 5 + 13 + 25 \; ....................... \; ( \,= 43 \,)$
$3^3 + 16^3 + 24^3 = 5^3 + 13^3 + 25^3 = 17947$

$5 + 15 + 23 = 8 + 11 + 24 \; ....................... \; ( \,= 43 \,)$
$5^3 + 15^3 + 23^3 = 8^3 + 11^3 + 24^3 = 15667$

$7 + 15 + 21 = 10 + 11 + 22 \; ...................... \; ( \,= 43 \,)$
$7^3 + 15^3 + 21^3 = 10^3 + 11^3 + 22^3 = 12979$

$1 + 20 + 25 = 7 + 11 + 28 \; ....................... \; ( \,= 46 \,)$
$1^3 + 20^3 + 25^3 = 7^3 + 11^3 + 28^3 = 23626$

$4 + 20 + 22 = 7 + 14 + 25 \; ....................... \; ( \,= 46 \,)$
$4^3 + 20^3 + 22^3 = 7^3 + 14^3 + 25^3 = 18712$

$5 + 17 + 25 = 7 + 14 + 26 \; ....................... \; ( \,= 47 \,)$
$5^3 + 17^3 + 25^3 = 7^3 + 14^3 + 26^3 = 20663$

$11 + 17 + 19 = 12 + 15 + 20 \; ..................... \; ( \,= 47 \,)$
$11^3 + 17^3 + 19^3 = 12^3 + 15^3 + 20^3 = 13103$

$3 + 16 + 29 = 8 + 10 + 30 \; ....................... \; ( \,= 48 \,)$
$3^3 + 16^3 + 29^3 = 8^3 + 10^3 + 30^3 = 28512$

$4 + 20 + 24 = 6 + 16 + 26 \; ....................... \; ( \,= 48 \,)$
$4^3 + 20^3 + 24^3 = 6^3 + 16^3 + 26^3 = 21888$

$8 + 18 + 22 = 9 + 16 + 23 \; ....................... \; ( \,= 48 \,)$
$8^3 + 18^3 + 22^3 = 9^3 + 16^3 + 23^3 = 16992$

$3 + 22 + 24 = 4 + 19 + 26 \; ....................... \; ( \,= 49 \,)$
$3^3 + 22^3 + 24^3 = 4^3 + 19^3 + 26^3 = 24499$

$4 + 17 + 28 = 7 + 13 + 29 \; ....................... \; ( \,= 49 \,)$
$4^3 + 17^3 + 28^3 = 7^3 + 13^3 + 29^3 = 26929$

$5 + 19 + 25 = 9 + 13 + 27 \; ....................... \; ( \,= 49 \,)$
$5^3 + 19^3 + 25^3 = 9^3 + 13^3 + 27^3 = 22609$

$7 + 19 + 23 = 10 + 14 + 25 \; ...................... \; ( \,= 49 \,)$
$7^3 + 19^3 + 23^3 = 10^3 + 14^3 + 25^3 = 19369$

$2 + 17 + 31 = 6 + 12 + 32 \; ....................... \; ( \,= 50 \,)$
$2^3 + 17^3 + 31^3 = 6^3 + 12^3 + 32^3 = 34712$

$2 + 20 + 28 = 6 + 14 + 30 \; ....................... \; ( \,= 50 \,)$
$2^3 + 20^3 + 28^3 = 6^3 + 14^3 + 30^3 = 29960$

$4 + 20 + 26 = 5 + 18 + 27 \; ....................... \; ( \,= 50 \,)$
$4^3 + 20^3 + 26^3 = 5^3 + 18^3 + 27^3 = 25640$

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 = 1,3

1. Paul says:

Here is a list with 40<= a + b + c<=50

$4 + 16 + 20 = 8 + 10 + 22 = 40$
$4^3 + 16^3 + 20^3 = 8^3 + 10^3 + 22^3 = 64000$

$1 + 17 + 23 = 5 + 11 + 25 = 41$
$1^3 + 17^3 + 23^3 = 5^3 + 11^3 + 25^3 = 68921$
$2 + 17 + 22 = 3 + 15 + 23 = 41$
$2^3 + 17^3 + 22^3 = 3^3 + 15^3 + 23^3 = 68921$
$5 + 16 + 20 = 6 + 14 + 21 = 41$
$5^3 + 16^3 + 20^3 = 6^3 + 14^3 + 21^3 = 68921$

$4 + 17 + 21 = 7 + 12 + 23 = 42$
$4^3 + 17^3 + 21^3 = 7^3 + 12^3 + 23^3 = 74088$
$4 + 18 + 20 = 9 + 10 + 23 = 42$
$4^3 + 18^3 + 20^3 = 9^3 + 10^3 + 23^3 = 74088$
$7 + 15 + 20 = 9 + 12 + 21 = 42$
$7^3 + 15^3 + 20^3 = 9^3 + 12^3 + 21^3 = 74088$

$1 + 19 + 23 = 3 + 15 + 25 = 43$
$1^3 + 19^3 + 23^3 = 3^3 + 15^3 + 25^3 = 79507$
$3 + 16 + 24 = 5 + 13 + 25 = 43$
$3^3 + 16^3 + 24^3 = 5^3 + 13^3 + 25^3 = 79507$
$5 + 15 + 23 = 8 + 11 + 24 = 43$
$5^3 + 15^3 + 23^3 = 8^3 + 11^3 + 24^3 = 79507$
$7 + 15 + 21 = 10 + 11 + 22 = 43$
$7^3 + 15^3 + 21^3 = 10^3 + 11^3 + 22^3 = 79507$

$1 + 20 + 25 = 7 + 11 + 28 = 46$
$1^3 + 20^3 + 25^3 = 7^3 + 11^3 + 28^3 = 97336$
$4 + 20 + 22 = 7 + 14 + 25 = 46$
$4^3 + 20^3 + 22^3 = 7^3 + 14^3 + 25^3 = 97336$

$5 + 17 + 25 = 7 + 14 + 26 = 47$
$5^3 + 17^3 + 25^3 = 7^3 + 14^3 + 26^3 = 103823$
$11 + 17 + 19 = 12 + 15 + 20 = 47$
$11^3 + 17^3 + 19^3 = 12^3 + 15^3 + 20^3 = 103823$

$3 + 16 + 29 = 8 + 10 + 30 = 48$
$3^3 + 16^3 + 29^3 = 8^3 + 10^3 + 30^3 = 110592$
$4 + 20 + 24 = 6 + 16 + 26 = 48$
$4^3 + 20^3 + 24^3 = 6^3 + 16^3 + 26^3 = 110592$
$8 + 18 + 22 = 9 + 16 + 23 = 48$
$8^3 + 18^3 + 22^3 = 9^3 + 16^3 + 23^3 = 110592$

$3 + 22 + 24 = 4 + 19 + 26 = 49$
$3^3 + 22^3 + 24^3 = 4^3 + 19^3 + 26^3 = 117649$
$4 + 17 + 28 = 7 + 13 + 29 = 49$
$4^3 + 17^3 + 28^3 = 7^3 + 13^3 + 29^3 = 117649$
$5 + 19 + 25 = 9 + 13 + 27 = 49$
$5^3 + 19^3 + 25^3 = 9^3 + 13^3 + 27^3 = 117649$
$7 + 19 + 23 = 10 + 14 + 25 = 49$
$7^3 + 19^3 + 23^3 = 10^3 + 14^3 + 25^3 = 117649$

$2 + 17 + 31 = 6 + 12 + 32 = 50$
$2^3 + 17^3 + 31^3 = 6^3 + 12^3 + 32^3 = 125000$
$2 + 20 + 28 = 6 + 14 + 30 = 50$
$2^3 + 20^3 + 28^3 = 6^3 + 14^3 + 30^3 = 125000$
$4 + 20 + 26 = 5 + 18 + 27 = 50$
$4^3 + 20^3 + 26^3 = 5^3 + 18^3 + 27^3 = 125000$

Paul.

• benvitalis says:

I posted your solutions. Check out the sums of cubes.