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multigrade




Sets of equations in which the sums of powers of two different sets of numbers are the same for several different exponents. The simplest example is:
1 + 6 + 8 = 2 + 4 + 9
12 + 62 + 82 = 22 + 42 + 92.
Another multigrade is:
1 + 8 + 10 + 17 = 36 = 2 + 5 + 13 + 16
12 + 82 + 102 + 172 = 454 = 22 + 52 + 132 + 162
13 + 83 + 103 + 173 = 6426 = 23 + 53 + 133 + 163.
Remarkably, if any integer is added to all the terms of a multigrade it will still hold. Adding 1 to the example above, gives the multigrade (2, 9, 11, 18); (3, 6, 14, 17) (n = 1,2,3).

Some high-order multigrades include:
(1, 50, 57, 15, 22, 71); (2, 45, 61, 11, 27, 70); (5, 37, 66, 6, 35, 67) (n = 1,2,3,4,5)
and
(1, 9, 25, 51, 75, 79, 107, 129, 131, 157, 159, 173); (3, 15, 19, 43, 89, 93, 97, 137, 139, 141, 167, 171) (n = 1, 3, 5, 7, 9, 11, 13).

Related category

   • NUMBER THEORY