Fermat's Last Theorem/Leonhard Euler

Leonhard Euler
$$\mathbb{T}$$he publication of Fermat’s writings had generated opposing opinions among mathematicians. The majority of them recognised the usefulness but the fact that the greater part of the theorems were without proof or with incomplete proofs obviously reduced the immediate usefulness of them even if some mathematicians took the theorems as challenges to face and win. Many were faced and resolved but that which subsequently would be called the last theorem resisted all attempted assaults. Leonhard Euler obtained the first results a century after Fermat. Euler was a Swiss mathematician born in 1707 in Basel and died in 1783 in St. Petersburg. Initially Euler was to have become a theologian but Johann Bernoulli became aware of the extraordinary ability of the young man and convinced his father to let Leonhard become a mathematician. This was an enormous good fortune for mathematics given that Euler’s contributions range over so many areas of mathematics and are so profound as to render Euler one of the greatest mathematicians of the XVIII century if not rightly the greatest. Euler analysing the notes written by Fermat found an outline proof of the case n=4. Fermat had written this proof within another proof. In order to prove that case Fermat made use of a technique called infinite descent, Euler sought to utilise this technique for the other cases in such a way as to find a proof for all values of n. Initially he confronted the case n=3. He succeeded in resolving this case but had to make use of complex numbers, in reality other mathematicians had sought to adapt infinite descent to the case n=3 but it took a creative person such as Euler in order to understand that complex numbers were necessary in order to obtain a valid proof. Euler also sought to resolve n=5 but without results.