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If there is a counterexample to Fermat's Last Theorem, then there is a counterexample $a^p+b^p=c^p$
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with $p$ an odd prime.
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\end{lemma}
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\begin{proof}\leanok
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Note: this proof is \href{https://leanprover-community.github.io/mathlib4_docs/Mathlib/NumberTheory/FLT/Four.html#FermatLastTheorem.of_odd_primes}{in mathlib already};
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we run through it for completeness' sake.
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Say $a^n + b^n = c^n$ is a counterexample to Fermat's Last Theorem. Every positive integer is either
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a power of 2 or has an odd prime factor. If $n=kp$ has an odd prime factor $p$ then
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$(a^k)^p+(b^k)^p=(c^k)^p$ is the counterexample we seek. It remains to deal with the case where
@@ -30,7 +33,7 @@ \section{Reduction to \texorpdfstring{$n\geq5$}{ngeq5} and prime}
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\begin{corollary}\label{WLOG_p_ge_5}\leanok If there is a counterexample to Fermat's Last Theorem, then there is a counterexample $a^p+b^p=c^p$ with $p$ prime and $p\geq5$.
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\end{corollary}
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\begin{proof}\uses{p_not_three, WLOG_n_prime}\leanok Follows from the previous two lemmas.\end{proof}
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\begin{proof}\uses{p_not_three, FermatLastTheorem.of_odd_primes}\leanok Follows from the previous two lemmas.\end{proof}
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