A^4 + B^4 + C^4 + D^4 + E^4 = N^4

 
 
    5^4 \; = \; 4^4 \; + \; 4^4 \; + \; 3^4 \; + \; 2^4 \; + \; 2^4

   60^4 \; = \; 52^4 \; + \; 48^4 \; + \; 24^4 \; + \; 8^4 \; + \; 8^4
   60^4 \; = \; 56^4 \; + \; 36^4 \; + \; 32^4 \; + \; 24^4 \; + \; 16^4

   65^4 \; = \; 63^4 \; + \; 36^4 \; + \; 24^4 \; + \; 16^4 \; + \; 12^4
   65^4 \; = \; 64^4 \; + \; 32^4 \; + \; 12^4 \; + \; 8^4 \; + \; 1^4
   65^4 \; = \; 52^4 \; + \; 46^4 \; + \; 44^4 \; + \; 39^4 \; + \; 2^4

   70^4 \; = \; 56^4 \; + \; 52^4 \; + \; 44^4 \; + \; 42^4 \; + \; 8^4

   75^4 \; = \; 65^4 \; + \; 60^4 \; + \; 30^4 \; + \; 10^4 \; + \; 10^4
   75^4 \; = \; 70^4 \; + \; 45^4 \; + \; 40^4 \; + \; 30^4 \; + \; 20^4

   80^4 \; = \; 64^4 \; + \; 64^4 \; + \; 48^4 \; + \; 32^4 \; + \; 32^4

   85^4 \; = \; 84^4 \; + \; 34^4 \; + \; 32^4 \; + \; 13^4 \; + \; 2^4
   85^4 \; = \; 84^4 \; + \; 38^4 \; + \; 22^4 \; + \; 16^4 \; + \; 13^4
   85^4 \; = \; 78^4 \; + \; 51^4 \; + \; 48^4 \; + \; 42^4 \; + \; 4^4
   85^4 \; = \; 73^4 \; + \; 68^4 \; + \; 38^4 \; + \; 24^4 \; + \; 8^4

   90^4 \; = \; 88^4 \; + \; 48^4 \; + \; 24^4 \; + \; 4^4 \; + \; 2^4
   90^4 \; = \; 76^4 \; + \; 72^4 \; + \; 48^4 \; + \; 16^4 \; + \; 2^4
   90^4 \; = \; 84^4 \; + \; 54^4 \; + \; 52^4 \; + \; 8^4 \; + \; 8^4
   90^4 \; = \; 78^4 \; + \; 72^4 \; + \; 36^4 \; + \; 12^4 \; + \; 12^4
   90^4 \; = \; 84^4 \; + \; 54^4 \; + \; 48^4 \; + \; 36^4 \; + \; 24^4

  130^4 \; = \; 126^4 \; + \; 72^4 \; + \; 48^4 \; + \; 32^4 \; + \; 24^4

  170^4 \; = \; 168^4 \; + \; 76^4 \; + \; 44^4 \; + \; 32^4 \; + \; 26^4
  170^4 \; = \; 168^4 \; + \; 68^4 \; + \; 64^4 \; + \; 26^4 \; + \; 4^4

  205^4 \; = \; 166^4 \; + \; 156^4 \; + \; 133^4 \; + \; 92^4 \; + \; 74^4
  205^4 \; = \; 156^4 \; + \; 146^4 \; + \; 142^4 \; + \; 133^4 \; + \; 4^4
  205^4 \; = \; 158^4 \; + \; 156^4 \; + \; 133^4 \; + \; 124^4 \; + \; 34^4
  205^4 \; = \; 164^4 \; + \; 156^4 \; + \; 133^4 \; + \; 106^4 \; + \; 58^4

  245^4 \; = \; 196^4 \; + \; 182^4 \; + \; 154^4 \; + \; 147^4 \; + \; 28^4
  245^4 \; = \; 196^4 \; + \; 188^4 \; + \; 147^4 \; + \; 142^4 \; + \; 46^4
  245^4 \; = \; 196^4 \; + \; 196^4 \; + \; 147^4 \; + \; 98^4 \; + \; 98^4

  250^4 \; = \; 200^4 \; + \; 200^4 \; + \; 150^4 \; + \; 100^4 \; + \; 100^4

  365^4 \; = \; 364^4 \; + \; 114^4 \; + \; 66^4 \; + \; 48^4 \; + \; 27^4
  365^4 \; = \; 364^4 \; + \; 102^4 \; + \; 96^4 \; + \; 27^4 \; + \; 6^4

  685^4 \; = \; 684^4 \; + \; 174^4 \; + \; 138^4 \; + \; 37^4 \; + \; 36^4
  685^4 \; = \; 684^4 \; + \; 162^4 \; + \; 156^4 \; + \; 37^4 \; + \; 6^4

1201^4 \; = \; 1200^4 \; + \; 260^4 \; + \; 220^4 \; + \; 40^4 \; + \; 49^4
1201^4 \; = \; 1200^4 \; + \; 280^4 \; + \; 140^4 \; + \; 140^4 \; + \; 49^4

 

Paul found:

   40^4 \; = \; 32^4 \; + \; 32^4 \; + \; 24^4 \; + \; 16^4 \; + \; 16^4

   45^4 \; = \; 36^4 \; + \; 36^4 \; + \; 27^4 \; + \; 18^4 \; + \; 18^4

   55^4 \; = \; 48^4 \; + \; 44^4 \; + \; 16^4 \; + \; 13^4 \; + \; 2^4
   55^4 \; = \; 52^4 \; + \; 33^4 \; + \; 28^4 \; + \; 14^4 \; + \; 2^4
   55^4 \; = \; 51^4 \; + \; 34^4 \; + \; 32^4 \; + \; 4^4 \; + \; 4^4
   55^4 \; = \; 46^4 \; + \; 44^4 \; + \; 31^4 \; + \; 6^4 \; + \; 4^4
   55^4 \; = \; 54^4 \; + \; 28^4 \; + \; 13^4 \; + \; 8^4 \; + \; 4^4

   62^4 \; = \; 60^4 \; + \; 34^4 \; + \; 20^4 \; + \; 20^4 \; + \; 20^4

   89^4 \; = \; 80^4 \; + \; 60^4 \; + \; 52^4 \; + \; 35^4 \; + \; 10^4

   93^4 \; = \; 90^4 \; + \; 51^4 \; + \; 30^4 \; + \; 30^4 \; + \; 30^4

   95^4 \; = \; 78^4 \; + \; 67^4 \; + \; 66^4 \; + \; 48^4 \; + \; 6^4
   95^4 \; = \; 76^4 \; + \; 74^4 \; + \; 57^4 \; + \; 52^4 \; + \; 22^4

 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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6 Responses to A^4 + B^4 + C^4 + D^4 + E^4 = N^4

  1. Nick Nash says:

    SImilar to Beal’s conjucture?

      • Nick Nash says:

        A^x + B^y =C^z
        A^x = D^w + E^u
        B^y = F^v + G^r
        So,
        D^w + E^u + F^v + G^r = C^z
        So, by Beal’s conjucture, (assuming is it right), B, F, and G have a GCD other than 1
        and A, D and E also have GCD other than 1.
        So, this can be expanded many times. Like D^w = L^p + N^q
        Thus, L^p+ N^q + E^u + F^v + G^r = C^z
        Where x, y, z, w, u, v, r, p, q > 2

  2. paul says:

    Here are all solutions with N <= 100

    2^4 + 2^4 + 3^4 + 4^4 + 4^4 = 5^4

    4^4 + 4^4 + 6^4 + 8^4 + 8^4 = 10^4

    2^4 + 2^4 + 6^4 + 12^4 + 13^4 = 15^4
    4^4 + 6^4 + 8^4 + 9^4 + 14^4 = 15^4
    6^4 + 6^4 + 9^4 + 12^4 + 12^4 = 15^4

    8^4 + 8^4 + 12^4 + 16^4 + 16^4 = 20^4

    10^4 + 10^4 + 15^4 + 20^4 + 20^4 = 25^4

    4^4 + 4^4 + 12^4 + 24^4 + 26^4 = 30^4
    8^4 + 12^4 + 16^4 + 18^4 + 28^4 = 30^4
    12^4 + 12^4 + 18^4 + 24^4 + 24^4 = 30^4

    10^4 + 10^4 + 10^4 + 17^4 + 30^4 = 31^4

    4^4 + 21^4 + 22^4 + 26^4 + 28^4 = 35^4
    14^4 + 14^4 + 21^4 + 28^4 + 28^4 = 35^4

    16^4 + 16^4 + 24^4 + 32^4 + 32^4 = 40^4

    1^4 + 2^4 + 12^4 + 24^4 + 44^4 = 45^4
    1^4 + 8^4 + 24^4 + 36^4 + 38^4 = 45^4
    4^4 + 4^4 + 26^4 + 27^4 + 42^4 = 45^4
    6^4 + 6^4 + 18^4 + 36^4 + 39^4 = 45^4
    12^4 + 18^4 + 24^4 + 27^4 + 42^4 = 45^4
    18^4 + 18^4 + 27^4 + 36^4 + 36^4 = 45^4

    20^4 + 20^4 + 30^4 + 40^4 + 40^4 = 50^4

    2^4 + 13^4 + 16^4 + 44^4 + 48^4 = 55^4
    2^4 + 14^4 + 28^4 + 33^4 + 52^4 = 55^4
    4^4 + 4^4 + 32^4 + 34^4 + 51^4 = 55^4
    4^4 + 6^4 + 31^4 + 44^4 + 46^4 = 55^4
    4^4 + 8^4 + 13^4 + 28^4 + 54^4 = 55^4
    22^4 + 22^4 + 33^4 + 44^4 + 44^4 = 55^4

    8^4 + 8^4 + 24^4 + 48^4 + 52^4 = 60^4
    16^4 + 24^4 + 32^4 + 36^4 + 56^4 = 60^4
    24^4 + 24^4 + 36^4 + 48^4 + 48^4 = 60^4

    20^4 + 20^4 + 20^4 + 34^4 + 60^4 = 62^4

    1^4 + 8^4 + 12^4 + 32^4 + 64^4 = 65^4
    2^4 + 39^4 + 44^4 + 46^4 + 52^4 = 65^4
    4^4 + 4^4 + 32^4 + 32^4 + 63^4 = 65^4
    12^4 + 16^4 + 24^4 + 36^4 + 63^4 = 65^4
    26^4 + 26^4 + 39^4 + 52^4 + 52^4 = 65^4

    8^4 + 42^4 + 44^4 + 52^4 + 56^4 = 70^4
    28^4 + 28^4 + 42^4 + 56^4 + 56^4 = 70^4

    10^4 + 10^4 + 30^4 + 60^4 + 65^4 = 75^4
    20^4 + 30^4 + 40^4 + 45^4 + 70^4 = 75^4
    30^4 + 30^4 + 45^4 + 60^4 + 60^4 = 75^4

    32^4 + 32^4 + 48^4 + 64^4 + 64^4 = 80^4

    2^4 + 13^4 + 32^4 + 34^4 + 84^4 = 85^4
    4^4 + 42^4 + 48^4 + 51^4 + 78^4 = 85^4
    8^4 + 24^4 + 38^4 + 68^4 + 73^4 = 85^4
    12^4 + 12^4 + 54^4 + 54^4 + 77^4 = 85^4
    13^4 + 16^4 + 22^4 + 38^4 + 84^4 = 85^4
    34^4 + 34^4 + 51^4 + 68^4 + 68^4 = 85^4

    10^4 + 35^4 + 52^4 + 60^4 + 80^4 = 89^4

    2^4 + 4^4 + 24^4 + 48^4 + 88^4 = 90^4
    2^4 + 16^4 + 48^4 + 72^4 + 76^4 = 90^4
    8^4 + 8^4 + 52^4 + 54^4 + 84^4 = 90^4
    12^4 + 12^4 + 36^4 + 72^4 + 78^4 = 90^4
    24^4 + 36^4 + 48^4 + 54^4 + 84^4 = 90^4
    36^4 + 36^4 + 54^4 + 72^4 + 72^4 = 90^4

    30^4 + 30^4 + 30^4 + 51^4 + 90^4 = 93^4

    6^4 + 48^4 + 66^4 + 67^4 + 78^4 = 95^4
    22^4 + 52^4 + 57^4 + 74^4 + 76^4 = 95^4
    38^4 + 38^4 + 57^4 + 76^4 + 76^4 = 95^4

    40^4 + 40^4 + 60^4 + 80^4 + 80^4 = 100^4

    Paul.

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