## Equal sums of 5-th powers (4 terms)

Here are a few with the terms less than 100

$0^5 + 2^5 + 37^5 + 50^5 = 9^5 + 11^5 + 17^5 + 52^5 = 381843989$
$0^5 + 3^5 + 22^5 + 25^5 = 1^5 + 8^5 + 14^5 + 27^5 = 14919500$
$1^5 + 10^5 + 27^5 + 32^5 = 8^5 + 9^5 + 19^5 + 34^5 = 48003340$
$1^5 + 13^5 + 17^5 + 23^5 = 3^5 + 9^5 + 21^5 + 21^5 = 8227494$
$3^5 + 38^5 + 41^5 + 56^5 = 13^5 + 31^5 + 36^5 + 58^5 = 745823388$
$3^5 + 41^5 + 42^5 + 54^5 = 9^5 + 35^5 + 40^5 + 56^5 = 705712700$
$4^5 + 11^5 + 13^5 + 22^5 = 5^5 + 6^5 + 19^5 + 20^5 = 5687000$
$4^5 + 13^5 + 26^5 + 31^5 = 11^5 + 14^5 + 16^5 + 33^5 = 40882844$
$5^5 + 20^5 + 21^5 + 24^5 = 9^5 + 17^5 + 18^5 + 26^5 = 15249850$
$6^5 + 43^5 + 49^5 + 54^5 = 8^5 + 39^5 + 52^5 + 53^5 = 888656492$
$7^5 + 14^5 + 33^5 + 36^5 = 11^5 + 12^5 + 29^5 + 38^5 = 100156200$
$7^5 + 16^5 + 72^5 + 89^5 = 9^5 + 12^5 + 76^5 + 87^5 = 7520042464$
$7^5 + 21^5 + 23^5 + 33^5 = 11^5 + 13^5 + 29^5 + 31^5 = 49672644$
$7^5 + 32^5 + 36^5 + 41^5 = 9^5 + 28^5 + 39^5 + 40^5 = 209893616$
$7^5 + 34^5 + 43^5 + 56^5 = 15^5 + 32^5 + 35^5 + 58^5 = 743192450$
$8^5 + 23^5 + 42^5 + 57^5 = 21^5 + 24^5 + 26^5 + 59^5 = 738852400$
$9^5 + 40^5 + 51^5 + 60^5 = 13^5 + 38^5 + 47^5 + 62^5 = 1225084300$

$12^5 + 21^5 + 23^5 + 38^5 = 13^5 + 14^5 + 31^5 + 36^5 = 90004444$
$12^5 + 32^5 + 36^5 + 48^5 = 14^5 + 26^5 + 42^5 + 46^5 = 349073408$
$13^5 + 39^5 + 59^5 + 71^5 = 15^5 + 35^5 + 63^5 + 69^5 = 2609749142$
$14^5 + 30^5 + 45^5 + 35^5 = 10^5 + 34^5 + 37^5 + 43^5 = 261887824$
$14^5 + 33^5 + 37^5 + 66^5 = 16^5 + 21^5 + 49^5 + 64^5 = 1361349750$
$16^5 + 29^5 + 41^5 + 74^5 = 17^5 + 18^5 + 53^5 + 72^5 = 2356422550$
$17^5 + 24^5 + 33^5 + 36^5 = 21^5 + 22^5 + 29^5 + 38^5 = 108984050$
$19^5 + 34^5 + 37^5 + 46^5 = 27^5 + 29^5 + 32^5 + 48^5 = 323218456$

Find other solutions

math grad - Interest: Number theory
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### One Response to Equal sums of 5-th powers (4 terms)

1. Paul says:

There are 1909 solutions (not all different) with a < b < c < d <=100, here are the first 5 and last 2 that are different terms. format is

{{a, b, c, d, etc},{e, f, g, h, etc}}

{{1,3,9,55,503343668},{15,22,38,53,503343668}}
{{1,3,28,53,435406105},{4,25,34,52,435406105}}
{{1,3,30,79,3101356643},{18,51,64,70,3101356643}}
{{1,3,52,91,6620525727},{8,16,31,92,6620525727}}
{{1,3,74,94,9558047092},{16,24,33,99,9558047092}}

….
{{53,79,85,91,14172626468},{60,68,87,93,14172626468}}
{{54,74,89,93,15219114790},{61,67,85,97,15219114790}}

Paul.