## a^4 + b^4 + c^4 = n^2

$a^4 \; + \; b^4 \; + \; c^4 \; = \; n^2$

a,   b,   c   are positive integers

$(a, b, c) = 1$

$12^4 + 15^4 + 20^4 \; = \; 481^2$
$60^4 + 65^4 + 156^4 \; = \; 24961^2$

$120^4 + 136^4 + 255^4 \; = \; 69121^2$
$168^4 + 175^4 + 600^4 \; = \; 362401^2$
$360^4 + 369^4 + 1640^4 \; = \; 2696161^2$
$420^4 + 580^4 + 609^4 \; = \; 530881^2$
$420^4 + 444^4 + 1295^4 \; = \; 1697761^2$
$660^4 + 671^4 + 3660^4 \; = \; 13410241^2$

$1008^4 + 1040^4 + 4095^4 \; = \; 16834561^2$
$1092^4 + 1105^4 + 7140^4 \; = \; 51008161^2$
$1680^4 + 1695^4 + 12656^4 \; = \; 160224961^2$
$1980^4 + 2020^4 + 9999^4 \; = \; 100140001^2$
$1260^4 + 1484^4 + 2385^4 \; = \; 6302881^2$
$1848^4 + 2145^4 + 3640^4 \; = \; 14435521^2$
$2448^4 + 2465^4 + 20880^4 \; = \; 436057921^2$
$2640^4 + 3504^4 + 4015^4 \; = \; 21428641^2$
$2772^4 + 3060^4 + 6545^4 \; = \; 44516641^2$
$3120^4 + 3471^4 + 7120^4 \; = \; 53007841^2$
$3420^4 + 3439^4 + 32580^4 \; = \; 1061586721^2$
$3432^4 + 3480^4 + 20735^4 \; = \; 430272001^2$
$4620^4 + 4641^4 + 48620^4 \; = \; 2364098881^2$
$4680^4 + 6305^4 + 6984^4 \; = \; 66626881^2$
$5148^4 + 5500^4 + 14625^4 \; = \; 217638721^2$
$5460^4 + 5516^4 + 38415^4 \; = \; 1476326881^2$
$5460^4 + 6540^4 + 9919^4 \; = \; 111346561^2$
$6072^4 + 6095^4 + 69960^4 \; = \; 4894681441^2$
$7140^4 + 7599^4 + 20860^4 \; = \; 441904801^2$
$8160^4 + 8224^4 + 65535^4 \; = \; 4295884801^2$
$8580^4 + 8996^4 + 28545^4 \; = \; 822128641^2$
$9240^4 + 12056^4 + 14385^4 \; = \; 266897761^2$

$10032^4 + 10545^4 + 32560^4 \; = \; 1070709601^2$
$11220^4 + 13345^4 + 20724^4 \; = \; 481684801^2$
$13260^4 + 13740^4 + 50609^4 \; = \; 2574230881^2$
$14280^4 + 20111^4 + 20280^4 \; = \; 611812321^2$
$15708^4 + 17220^4 + 38335^4 \; = \; 1519359361^2$
$15912^4 + 19240^4 + 28305^4 \; = \; 918158881^2$
$15960^4 + 18335^4 + 32424^4 \; = \; 1132766401^2$
$17940^4 + 18561^4 + 69940^4 \; = \; 4914270721^2$
$19380^4 + 19924^4 + 83505^4 \; = \; 6994466401^2$
$20748^4 + 27265^4 + 31980^4 \; = \; 1335621121^2$
$21840^4 + 24465^4 + 48464^4 \; = \; 2470309921^2$
$23712^4 + 25440^4 + 65455^4 \; = \; 4369291681^2$
$23940^4 + 30940^4 + 37791^4 \; = \; 1812319681^2$
$25080^4 + 28920^4 + 50369^4 \; = \; 2744396161^2$
$28980^4 + 31855^4 + 69804^4 \; = \; 5047499041^2$
$36960^4 + 44960^4 + 64911^4 \; = \; 4868797921^2$
$38640^4 + 46529^4 + 69360^4 \; = \; 5482707841^2$

math grad - Interest: Number theory
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### 2 Responses to a^4 + b^4 + c^4 = n^2

1. paul says:

I have extended and added to the list, and even now I am not 100% that they are all accounted for within any given range for a. the list is too big for here so here is a link to the dropbox where “a” has up to 9 digits, (not all 9 digit ones are there for sure)

https://www.dropbox.com/s/0g3inm0d3cwf4bc/sum%20of%20fourth%20powers.txt?dl=0

Paul.

• benvitalis says:

I’m not sure either. Thanks for extending the list.