a^3 + b^3 + c^3 + d^3 = n^3 and a+d = b+c

 
 
a \; + \; d \; = \; b \; + \; c,    and

a^3 \; + \; b^3 \; + \; c^3 \; + \; d^3 \; = \; n^3

 
For example,

 

9 \; + 23 \; = \; 13 \; + \; 19 \; = \; 32

9^3 \; + \; 13^3 \; + \; 19^3 \; + \; 23^3 \; = \; 21952 \; = \; 28^3

 

11 \; + \; 14 \; = \; 12 \; + \; 13 \; = \; 25 \; = \; 5^2

11^3 \; + \; 12^3 \; + \; 13^3 \; + \; 14^3 \; = \; 20^3

The multiples of   [11, 12, 13, 14]

22 + 28 = 24 + 26 = 50 \; ....... \; 22^3 + 28^3 + 24^3 + 26^3 = 40^3
33 + 42 = 36 + 39 = 75 \; ....... \; 33^3 + 42^3 + 36^3 + 39^3 = 60^3
44 + 56 = 48 + 52 = 100 \; ...... \; 44^3 + 56^3 + 48^3 + 52^3 = 80^3
55 + 70 = 60 + 65 = 125 \; ...... \; 55^3 + 70^3 + 60^3 + 65^3 = 100^3
66 + 84 = 72 + 78 = 150 \; ...... \; 66^3 + 84^3 + 72^3 + 78^3 = 120^3
77 + 98 = 84 + 91 = 175 \; ...... \; 77^3 + 98^3 + 84^3 + 91^3 = 140^3
………………………………………………
………………………………………………

 
The multiples of   [9, 13, 19, 23]

18 + 46 = 26 + 38 = 64 \; ....... \; 18^3 + 26^3 + 38^3 + 46^3 = 56^3
27 + 69 = 39 + 57 = 96 \; ....... \; 27^3 + 39^3 + 57^3 + 69^3 = 84^3
36 + 92 = 52 + 76 = 128 \; ...... \; 36^3 + 52^3 + 76^3 + 92^3 = 112^3
45 + 115 = 65 + 95 = 160 \; ..... \; 45^3 + 65^3 + 95^3 + 115^3  = 140^3
54 + 138 = 78 + 114 = 192 \; .... \; 54^3 + 78^3 + 114^3 + 138^3 = 168^3
63 + 161 = 91 + 133 = 224 \; .... \; 63^3 + 91^3 + 133^3 + 161^3 = 196^3
72 + 184 = 104 + 152 = 256 \; ... \; 72^3 + 104^3 + 152^3 + 184^3 = 224^3
…………………………………………………….
…………………………………………………….

 
13 + 23 = 15 + 21 = 36 \; ..... \; 13^3 + 15^3 + 21^3 + 23^3 = 30^3

The multiples of   [13, 15, 21, 23]

26 + 46 = 30 + 42 = 72 \; ...... \; 26^3 + 30^3 + 42^3 + 46^3 = 60^3
39 + 69 = 45 + 63 = 108 \; ..... \; 39^3 + 45^3 + 63^3 + 69^3 = 90^3
52 + 92 = 60 + 84 = 144 \; ..... \; 52^3 + 60^3 + 84^3 + 92^3 = 120^3
65 + 115 = 75 + 105 = 180 \; ... \; 65^3 + 75^3 + 105^3 + 115^3 = 150^3
………………………….
…………………………..

 
Paul found :

9 + 23 = 13 + 19 = 32 \; ..... \; 9^3 + 13^3 + 19^3 + 23^3 = 28^3
9 + 91 = 17 + 83 = 100 \; ..... \; 9^3 + 17^3 + 83^3 + 91^3 = 110^3
56 + 69 = 58 + 67 = 125 \; .... \; 56^3 + 58^3 + 67^3 + 69^3  = 100^3
43 + 117 = 71 + 89 = 160 \; ..... \; 43^3 + 71^3 + 89^3 + 117^3 = 140^3
47 + 113 = 61 + 99 = 160 \; ..... \; 47^3 + 61^3 + 99^3 + 113^3 = 140^3
4 + 165 = 59 + 110 = 169 \; ..... \; 4^3 + 59^3 + 110^3 + 165^3 = 182^3
20 + 149 = 30 + 139 = 169 \; ..... \; 20^3 + 30^3 + 139^3 + 149^3 = 182^3
61 + 119 = 87 + 93 = 180 \; ..... \; 61^3 + 87^3 + 93^3 + 119^3 = 150^3
63 + 117 = 79 + 101 = 180 \; ..... \; 63^3 + 79^3 + 101^3 + 117^3 = 150^3
18 + 182 = 34 + 166 = 200 \; ..... \; 18^3 + 182^3 + 34^3 + 166^3 = 220^3
31 + 169 = 98 + 102 = 200 \; ..... \; 31^3 + 169^3 + 98^3 + 102^3 = 190^3
46 + 154 = 57 + 143 = 200 \; ..... \; 46^3 + 57^3 + 143^3 + 154^3 = 190^3

                               ————————————————–          

 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

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

2 Responses to a^3 + b^3 + c^3 + d^3 = n^3 and a+d = b+c

  1. paul says:

    Here are a few with the sum of a & b <=200

    13 + 23 = 15 + 21 = 36
    13^3 + 23^3 + 15^3 + 21^3 = 27000 = 30^3

    22 + 28 = 24 + 26 = 50
    22^3 + 28^3 + 24^3 + 26^3 = 64000 = 40^3

    18 + 46 = 26 + 38 = 64
    18^3 + 46^3 + 26^3 + 38^3 = 175616 = 56^3

    26 + 46 = 30 + 42 = 72
    26^3 + 46^3 + 30^3 + 42^3 = 216000 = 60^3

    33 + 42 = 36 + 39 = 75
    33^3 + 42^3 + 36^3 + 39^3 = 216000 = 60^3

    27 + 69 = 39 + 57 = 96
    27^3 + 69^3 + 39^3 + 57^3 = 592704 = 84^3

    9 + 91 = 17 + 83 = 100
    9^3 + 91^3 + 17^3 + 83^3 = 1331000 = 110^3
    44 + 56 = 48 + 52 = 100
    44^3 + 56^3 + 48^3 + 52^3 = 512000 = 80^3

    39 + 69 = 45 + 63 = 108
    39^3 + 69^3 + 45^3 + 63^3 = 729000 = 90^3

    55 + 70 = 60 + 65 = 125
    55^3 + 70^3 + 60^3 + 65^3 = 1000000 = 100^3
    56 + 69 = 58 + 67 = 125
    56^3 + 69^3 + 58^3 + 67^3 = 1000000 = 100^3

    36 + 92 = 52 + 76 = 128
    36^3 + 92^3 + 52^3 + 76^3 = 1404928 = 112^3

    52 + 92 = 60 + 84 = 144
    52^3 + 92^3 + 60^3 + 84^3 = 1728000 = 120^3

    66 + 84 = 72 + 78 = 150
    66^3 + 84^3 + 72^3 + 78^3 = 1728000 = 120^3

    43 + 117 = 71 + 89 = 160
    43^3 + 117^3 + 71^3 + 89^3 = 2744000 = 140^3
    45 + 115 = 65 + 95 = 160
    45^3 + 115^3 + 65^3 + 95^3 = 2744000 = 140^3
    47 + 113 = 61 + 99 = 160
    47^3 + 113^3 + 61^3 + 99^3 = 2744000 = 140^3

    4 + 165 = 59 + 110 = 169
    4^3 + 165^3 + 59^3 + 110^3 = 6028568 = 182^3
    20 + 149 = 30 + 139 = 169
    20^3 + 149^3 + 30^3 + 139^3 = 6028568 = 182^3

    77 + 98 = 84 + 91 = 175
    77^3 + 98^3 + 84^3 + 91^3 = 2744000 = 140^3

    61 + 119 = 87 + 93 = 180
    61^3 + 119^3 + 87^3 + 93^3 = 3375000 = 150^3
    63 + 117 = 79 + 101 = 180
    63^3 + 117^3 + 79^3 + 101^3 = 3375000 = 150^3
    65 + 115 = 75 + 105 = 180
    65^3 + 115^3 + 75^3 + 105^3 = 3375000 = 150^3

    54 + 138 = 78 + 114 = 192
    54^3 + 138^3 + 78^3 + 114^3 = 4741632 = 168^3

    18 + 182 = 34 + 166 = 200
    18^3 + 182^3 + 34^3 + 166^3 = 10648000 = 220^3
    31 + 169 = 98 + 102 = 200
    31^3 + 169^3 + 98^3 + 102^3 = 6859000 = 190^3
    46 + 154 = 57 + 143 = 200
    46^3 + 154^3 + 57^3 + 143^3 = 6859000 = 190^3
    88 + 112 = 96 + 104 = 200
    88^3 + 112^3 + 96^3 + 104^3 = 4096000 = 160^3

    There are a few multiples amongst those

    Paul.

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