Note for the Beal's Conjecture

This work explores two famous conjectures in number theory: Fermat's Last Theorem and Beal's Conjecture. Fermat's Last Theorem, posed by Pierre de Fermat in the 17th century, states that there are no positive integer solutions for the equation aⁿ + bⁿ = cⁿ, where n is greater than $2$. This theorem remained unproven for centuries until Andrew Wiles published a proof in 1994. Beal's Conjecture, formulated in 1997 by Andrew Beal, generalizes Fermat's Last Theorem. It states that for positive integers A, B, C, x, y, and z, if Aˣ + Bʸ = Cᶻ (where x, y, and z are all greater than $2$), then A, B, and C must share a common prime factor. Beal's Conjecture remains unproven, and a significant prize is offered for a solution. This paper provides a concise introduction to both conjectures, highlighting their connection and the ongoing challenge that a short proof for the Beal's Conjecture presents to mathematicians.