Competition problem — Number theory

Number theory Difficulty 4.8 AIME Prove it Brazil

Show that there are infinitely many positive integer solutions to $a^3 + 1990b^3 = c^4$ .

Solution

Take, for instance, $a = b = 1991k^4$ and $c = 1991k^3$ (there are lots of other solutions; can you find them all?).

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