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In Fermat's lifetime he proved the case when n=4
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Fermat's initial conjecture of Fermat's Last Theorem
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Euler proved Fermat's Last Theorem for the case n=3
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Lejeune Dirichlet proved Fermat's Last Theorem for the case n=5
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Adrien Marie Legendre proved Fermat's Last Theorem for the case n=5
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Gabriel Lame proved Fermat's Last Theorem for the case n=7
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With the use of a computer, Fermat's Last Theorem was proved for all n < 125,000
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Yoichi Miyaoka proved Fermat's Last Theorem in an arithmetic algebraic geometric way which made this proof easier for the public to understand. There were flaws in this proof.
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Finally in 1995 a valid proof of Fermat's Last Theorem was presented and publish in the Annals of Mathematics