A Note on Fermat's Equation

Around $1637$, Pierre de Fermat famously scribbled, and claimed to have a proof for, his statement that equation $a^{n} + b^{n} = c^{n}$ has no positive integer solutions for exponents $n>2$. The theorem stood unproven for centuries until Andrew Wiles' groundbreaking work in $1994$, with a notable caveat: Wiles' proof, while successful, relied on modern tools far beyond Fermat's claimed approach in terms of complexity. The present work potentially offers a solution which is closer in spirit to Fermat's original idea. The same tools designed to this effect are then used to prove the Beal conjecture, a well-known generalization of Fermat's Last Theorem.