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A Radical Proof

Problem

The radical of $n$, $\text{rad}(n)$, is the product of distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\text{rad}(504) = 2 \times 3 \times 7 = 42$.

Given any triplet of relatively prime positive integers $(a, b, c)$ for which $a + b = c$ and with $a \lt b \lt c$, it is conjectured, but not yet proved, that the largest element of the triplet, $c \lt \text{rad}(abc)^2$.

Assuming that this conjecture is true, prove that $x^n + y^n = z^n$ has no integer solutions for $n \ge 6$.

Problem ID: 284 (23 Jul 2006)     Difficulty: 3 Star

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