To make {3(4m^4 – n^4), 8(9m^4 – n^4), 15(16m^4 – n^4), 24(25m^4 – n^4)} squares

 
 
To make each of the following expression squares:

3 \, (4 \, m^4 \; - \; n^4)
8 \, (9 \, m^4 \; - \; n^4)
15 \, (16 \, m^4 \; - \; n^4)
24 \, (25 \, m^4 \; - \; n^4)

m   and   n   are positive integers with   m \; \neq \; n

 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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2 Responses to To make {3(4m^4 – n^4), 8(9m^4 – n^4), 15(16m^4 – n^4), 24(25m^4 – n^4)} squares

  1. paul says:

    Each one can individually be made square for some m and n, I think it impossible to make them simultaneously square for the same m and n. they each follow the same recurrence for m and n with {2, -1} and the resulting square is also the same recurrence for each with {3, -3, 1}. Here are the first 10 values for each

    {23,37,4653}
    {46,74,18612}
    {69,111,41877}
    {92,148,74448}
    {115,185,116325}
    {138,222,167508}
    {161,259,227997}
    {184,296,297792}
    {207,333,376893}
    {230,370,465300}

    3(4 37^4 – 23^4) = 4653^2
    3(4 74^4 – 46^4) = 18612^2
    3(4 111^4 – 69^4) = 41877^2
    3(4 148^4 – 92^4) = 74448^2
    3(4 185^4 – 115^4) = 116325^2
    3(4 222^4 – 138^4) = 167508^2
    3(4 259^4 – 161^4) = 227997^2
    3(4 296^4 – 184^4) = 297792^2
    3(4 333^4 – 207^4) = 376893^2
    3(4 370^4 – 230^4) = 465300^2

    {47,33,6808}
    {94,66,27232}
    {141,99,61272}
    {188,132,108928}
    {235,165,170200}
    {282,198,245088}
    {329,231,333592}
    {376,264,435712}
    {423,297,551448}
    {470,330,680800}

    8(9 33^4 – 47^4) = 6808^2
    8(9 66^4 – 94^4) = 27232^2
    8(9 99^4 – 141^4) = 61272^2
    8(9 132^4 – 188^4) = 108928^2
    8(9 165^4 – 235^4) = 170200^2
    8(9 198^4 – 282^4) = 245088^2
    8(9 231^4 – 329^4) = 333592^2
    8(9 264^4 – 376^4) = 435712^2
    8(9 297^4 – 423^4) = 551448^2
    8(9 330^4 – 470^4) = 680800^2

    {14,23,8160}
    {28,46,32640}
    {42,69,73440}
    {56,92,130560}
    {70,115,204000}
    {84,138,293760}
    {98,161,399840}
    {112,184,522240}
    {126,207,660960}
    {140,230,816000}

    15(16 23^4 – 14^4) = 8160^2
    15(16 46^4 – 28^4) = 32640^2
    15(16 69^4 – 42^4) = 73440^2
    15(16 92^4 – 56^4) = 130560^2
    15(16 115^4 – 70^4) = 204000^2
    15(16 138^4 – 84^4) = 293760^2
    15(16 161^4 – 98^4) = 399840^2
    15(16 184^4 – 112^4) = 522240^2
    15(16 207^4 – 126^4) = 660960^2
    15(16 230^4 – 140^4) = 816000^2

    {431,193,65736}
    {862,386,262944}
    {1293,579,591624}
    {1724,772,1051776}
    {2155,965,1643400}
    {2586,1158,2366496}
    {3017,1351,3221064}
    {3448,1544,4207104}
    {3879,1737,5324616}
    {4310,1930,6573600}

    24(25 193^4 – 431^4) = 65736^2
    24(25 386^4 – 862^4) = 262944^2
    24(25 579^4 – 1293^4) = 591624^2
    24(25 772^4 – 1724^4) = 1051776^2
    24(25 965^4 – 2155^4) = 1643400^2
    24(25 1158^4 – 2586^4) = 2366496^2
    24(25 1351^4 – 3017^4) = 3221064^2
    24(25 1544^4 – 3448^4) = 4207104^2
    24(25 1737^4 – 3879^4) = 5324616^2
    24(25 1930^4 – 4310^4) = 6573600^2

    Paul.

    • benvitalis says:

      So it seems that we can only obtain
      {23,37,4653},{47,33,6808},{14,23,8160},{431,193,65736} and their multiples.
      Yes, we cannot make them simultaneously square for the same m and n
      Not so interesting after all.

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