## A^2 + B^2 + C^2 = D^2

Consider the identity

$((ac)^2-(bd)^2)^2 + (2 \, abcd)^2 + ((ad)^2-(bc)^2)^2 \; = \; (a^4 + b^4) \, (c^4 + d^4)$

If    $a^4 + b^4 \; = \; c^4 + d^4$

then we have a square that can be expressed as the sum of three squares.

Here’s a solution:

$(a, \; b, \; c, \; d) \; = \; (133, \; 134, \; 59, \; 158)$

$386678175^2 + 332273368^2 + 379083360^2 \; = \; 635318657^2$
$= \; (133^4 + 134^4)(59^4 + 158^4)$

Find the next few solutions

Paul found some primitive and multiples solutions;

–>   the primitive solutions:

$\{a, \; b, \; c, \; d \} \; = \; \{157, \; 227, \; 7, \; 239 \}$
$157^4 \; + \; 227^4 \; = \; 7^4 \; + \; 239^4 \; = \; 3262811042$
$2942180208^2 \; + \; 119248094^2 \; + \; 1405450608^2 \; = \; 3262811042^2$

$\{a, \; b, \; c, \; d \} \; = \; \{256, \; 257, \; 193, \; 292 \}$
$256^4 \; + \; 257^4 \; = \; 193^4 \; + \; 292^4 \; = \; 8657437697$
$3190451472^2 \; + \; 7415547904^2 \; + \; 3127602303^2 \; = \; 8657437697^2$

$\{a, \; b, \; c, \; d \} \; = \; \{359, \; 514, \; 103, \; 542 \}$
$359^4 \; + \; 514^4 \; = \; 103^4 \; + \; 542^4 \; = \; 86409838577$
$76243975215^2 \; + \; 20602696952^2 \; + \; 35057742720^2 \; = \; 86409838577^2$

$\{a, \; b, \; c, \; d \} \; = \; \{503, 558, 222, 631 \}$
$503^4 \; + \; 558^4 \; = \; 222^4 \; + \; 631^4 \; = \; 160961094577$
$111503706048^2 \; + \; 78634750536^2 \; + \; 85393053073^2 \; = \; 160961094577^2$

$\{a, \; b, \; c, \; d \} \; = \; \{298, 497, 271, 502 \}$
$298^4 \; + \; 497^4 \; = \; 271^4 \; + \; 502^4 \; = \; 68899596497$
$55725401472^2 \; + \; 40297272904^2 \; + \; 4238375247^2 \; = \; 68899596497^2$

–>   the multiples ones:

$\{314, \; 454, \; 14, \; 478 \} \; = \; 2 \times \{157, \; 227, \; 7, \; 239 \}$
$\{266, \; 268, \; 118, \; 316 \} \; = \; 2 \times \{133, \; 134, \; 59, \; 158 \}$
$\{399, \; 402, \; 177, \; 474 \} \; = \; 3 \times \{133, \; 134, \; 59, \; 158 \}$
$\{471, \; 681, \; 21, \; 717 \} \; = \; 3 \times \{157, \; 227, \; 7, \; 239 \}$
$\{628, \; 908, \; 28, \; 956 \} \; = \; 4 \times \{157, \; 227, \; 7, \; 239 \}$
$\{532, \; 536, \; 236, \; 632 \} \; = \; 4 \times \{133, \; 134, \; 59, \; 158 \}$
$\{665, \; 670, \; 295, \; 790 \} \; = \; 5 \times \{133, \; 134, \; 59, \; 158 \}$
$\{798, \; 804, \; 354, \; 948 \} \; = \; 6 \times \{133, \; 134, \; 59, \; 158 \}$
$\{512, \; 514, \; 386, \; 584 \} \; = \; 2 \times \{256, \; 257, \; 193, \; 292 \}$
$\{768, \; 771, \; 579, \; 876 \} \; = \; 3 \times \{256, \; 257, \; 193, \; 292 \}$

math grad - Interest: Number theory
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### 3 Responses to A^2 + B^2 + C^2 = D^2

1. Paul says:

Here are a few more

{a,b,c,d} = {157,227,7,239}
2942180208^2 + 119248094^2 + 1405450608^2 = 3262811042^2

{a,b,c,d} = {314,454,14,478}
47074883328^2 + 1907969504^2 + 22487209728^2 = 52204976672^2

{a,b,c,d} = {266,268,118,316}
6186850800^2 + 5316373888^2 + 6065333760^2 = 10165098512^2

{a,b,c,d} = {399,402,177,474}
31320932175^2 + 26914142808^2 + 30705752160^2 = 51460811217^2

{a,b,c,d} = {256,257,193,292}
3190451472^2 + 7415547904^2 + 3127602303^2 = 8657437697^2

Paul.

2. Paul says:

and a few more

{a,b,c,d} = {471,681,21,717}
238316596848^2 + 9659095614^2 + 113841499248^2 = 264287694402^2

{a,b,c,d} = {628,908,28,956}
753198133248^2 + 30527512064^2 + 359795355648^2 = 835279626752^2

{a,b,c,d} = {359,514,103,542}
76243975215^2 + 20602696952^2 + 35057742720^2 = 86409838577^2

{a,b,c,d} = {503,558,222,631}
111503706048^2 + 78634750536^2 + 85393053073^2 = 160961094577^2

{a,b,c,d} = {532,536,236,632}
98989612800^2 + 85061982208^2 + 97045340160^2 = 162641576192^2

{a,b,c,d} = {298,497,271,502}
55725401472^2 + 40297272904^2 + 4238375247^2 = 68899596497^2

{a,b,c,d} = {665,670,295,790}
241673859375^2 + 207670855000^2 + 236927100000^2 = 397074160625^2

{a,b,c,d} = {798,804,354,948}
501134914800^2 + 430626284928^2 + 491292034560^2 = 823372979472^2

{a,b,c,d} = {512,514,386,584}
51047223552^2 + 118648766464^2 + 50041636848^2 = 138519003152^2

{a,b,c,d} = {768,771,579,876}
258426569232^2 + 600659380224^2 + 253335786543^2 = 701252453457^2

Paul.

• benvitalis says:

I posted your solutions (primitives and multiples)