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 \}

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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.

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