A, B and C are Positive integers, Can You Solve? | Nice Algebra Problem | Math Olympiad Problem

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This Problem is amazing So I dare you to solve!
What are the three distinct positive integers a, b, and c, such that a^3 + b^3 = c^4 ?
This is nice algebra problem from Math Olympiad
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I think this solution isn't exacte because every c existe in Z+ don't attein this expression k^3+l^3 . Thank you

dianadiana
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Nice but do you find ALL solutions? I think you only find solutions where c is the sum of two cubics. But could there also be solutions where c is not a sum of two cubics?

klausg