On Christmas Day in 1640, Pierre de Fermat sent a letter to a fellow mathematician famously claiming that every prime number of the form 4k+1 can be uniquely expressed as the sum of two integers. This conjecture was unsolved until Euler proved it over a century later; much time after, many mathematicians would come up with their own iterations and versions of proving the theorem. In this presentation, we will review and go into detail with some of the most popular proofs on this theorem, which will include Euler's proof of infinite descent, Zagier's "one-sentence proof," and the Gaussian Integer proof.
