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Let a, b, and c be positive integers such that a is the cube of an integer, c = b + 1, and a^2 + b^2 = c^2.  Find the least possible value of c.

 Dec 8, 2019
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\(a^2 + b^2 = c^2\\ a^2 + b^2 = (b + 1)^2\\ 2b + 1 = a^2\\ \text{Let }a = k^3,\\ k^6 - 1 = 2b\\ k^6 - 1 \equiv 0 \pmod 2\implies k\equiv 1 \pmod 2\\ \text{Exhaust }k.\\ \text{When }k = 1, b = 0\text{(rejected)}\\ \text{When }k = 3, b = 364\\ c = 364 + 1 = \boxed{365}\\ \text{Check:}\\ a^2 + 364^2 = 365^2\\ a = 729 = 9^3\)

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 Dec 8, 2019

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