Number Theory

A Complete Understanding of the Beal Conjecture

Authors: Felipe A. Wescoup

This paper presents a proof of the Beal Conjecture by means of a geometric and set-theoretic argument. Each term in the conjecture — A^x, B^y, and C^z — is interpreted as a volume composed of three-dimensional seed cubes whose side lengths are determined by the prime factors of the base. From this foundation, the prime factor sets of A and B are shown to be either disjoint or overlapping. The disjoint case is eliminated by contradiction: if A and B share no common prime, the equation A^x + B^y = C^z cannot be satisfied for any integer C and exponent z ≥ 3. The overlapping case directly implies that A, B, and C share a common prime factor, which is precisely what the conjecture asserts. Because these two cases are exhaustive, the conjecture is proven.

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[v1] 2026-04-27 23:44:21

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