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People will tell you that 1729 is the smallest number that is a sum of two integer cubes in two different ways, but they are mistaken. Actually the smallest such number is 91: $$91 = 3^3 + 4^3 = 6^3 + (5)^3 $$ 1729 is the smallest number that is a sum of two positive integer cubes in two different ways. (!!1729 = 1^3 + 12^3 = 9^3 + 10^3.!!) People sometimes forget that cubes, unlike squares, can be negative.
