A Note on the Beal Conjecture

In 1993, Andrew Beal formulated the conjecture that if A^x + B^y = C^z holds for positive integers A, B, C, x, y, z with x, y, z > 2, then A, B, and C must share a common prime factor. We prove the Beal conjecture for every primitive solution in which C possesses at least one odd prime divisor, using three classical tools: the Lifting The Exponent Lemma for odd primes, the exact p-adic valuation of the binomial coefficient (p^m _k ), and a careful pairing of the interior terms of the binomial expansion. The argument raises the equation to the power p^m (where m*p^m= ν_p(C^z)) and shows that the resulting righthand side has p-adic valuation exactly 2m, while the left-hand side has p-adic valuation m · p^m > 2m, yielding the required contradiction. The residual case in which C is a power of two is left open as a conjecture. For Fermat’s Last Theorem, however, the case c = 2^s admits a complete elementary proof: when the exponent n is even, an arithmetic argument modulo 8 gives ν₂(aⁿ + bⁿ) = 1 ̸= sn; when n is odd, the Lifting The Exponent Lemma for p = 2 forces a + b ≥ 2^sn, contradicting the strict bound a + b ≤ 2^s+1 − 2. Combining both results yields a complete proof of Fermat’s Last Theorem for all exponents n ≥ 3.