Mixed power relations -3
Sum of three fourth powers where the greatest number is the sum of the two other numbers is always associated with a square .
a4 + b4 + c4 = 2 d2 where a + b = c
If d is a square number (d = e2) it can also be expressed as a4 + b4 + c4 = 2 e4. This is exemplified with a typical example
14 + 74 + 84 = 6498 = 2 x 572
24 + 64 + 84 = 5408 = 2 x 522 = 25 x 132
34 + 54 + 84 = 4802 = 2 x 492 = 2 x 74
44 + 44 + 84 = 4608 = 2 x 482 = 2 x 122
Some other examples are
74 + 84 + 154 = 2 x 134
54 + 164 + 214 = 2 x 192
94 + 154 + 244 = 2 x 212
114 + 244 + 354 = 2 x 312
144 + 164 + 304 = 2 x 262
214 + 304 + 564 = 2 x 494
Another interesting property of this relation a4 + b4 + c4 = 2 d2 where a + b = c is
a2 + b2 + c2 = 2 d
12 + 72 + 82 = 114 = 2 x 57
22 + 62 + 82 = 2 x 52
It gives.
2(a4 + b4 + c4 ) = (2d)2 = (a2 + b2 + c2)2
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