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Something special about the number 1728

Someone said that 1,728 is the lowest integer that is the sum of two cube numbers in two different ways. Prove it

Looking for cube number sums , excluding the cube 1 = 1x1x1


64 = Cube product (8 x 8)
216 = Cube product (8 x 27)
512 = Cube product (8 x 64)
729 = Cube product (27 x 27)
1000 = Cube product (8 x 125)
1728 = Cube product (8 x 216) = Cube product (27 x 64)
So 1000 disproves the original theory, which needs to be modified so exclude the cube number 1 ( 1x1x1) 1000 = (10x10x10) x (1x1x1) or (2x2x2 = 8 ) x (5x5x5 = 125) If we qualify the original theory to exclude the first cube number 1, the theory is correct.
Can you write small computer programs with loops to investigate number theories?
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