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In 13 Lectures on Fermat's Last Theorem, Ribenboim states the following theorem (on page 7) attributed to Cauchy: If the first case of Fermat's theorem fails for the exponent $p$, then the sum: $$ 1^{...

As many of you may know, the illustrious L. Euler put forward a proof of the case $n=3$ of Fermat's Last Theorem via infinite descent. The thing is that, at a certain point, he resorted to the ...

One of the forms in which the Diophantine equation in question can be found in the literature is this: Solve the equation \begin{eqnarray}z^{2} = 4xy-x-y \qquad \qquad (\ast)\end{eqnarray} in ...

Fermat proved that $x^3-y^2=2$ has only one solution $(x,y)=(3,5)$. After some search, I only found proofs using factorization over the ring $Z[\sqrt{-2}]$. My question is: Is this Fermat's original ...