Beal's conjecture

Beal's conjecture states that if positive integers satisfy A^x + B^y = C^z with x, y, z greater than 2, then A, B, and C share a prime factor. It remains unproved.

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Perfect power: An integer greater than one that can be written as an integer power with exponent greater than one. Each term in the conjectured equation is a perfect power.
Fermat–Catalan conjecture: The conjecture that only finitely many coprime positive-integer solutions exist to a^p + b^q = c^r when 1/p + 1/q + 1/r is less than one. Its known solutions include the pairwise-coprime cases that constrain Beal's conjecture.
Coprime integers: Integers whose greatest common divisor is one. A counterexample requires the three bases to be pairwise coprime.
ABC conjecture: A conjecture relating the radical of a+b to the size of coprime positive integers a and b satisfying a+b=c. Its proposed inequalities imply strong restrictions on large coprime power equations.
Prime number: An integer greater than one whose only positive divisors are one and itself. The conclusion concerns a prime dividing all three bases.
2^3 + 1^7 = 3^4: The integer identity 8 + 1 = 81 is false. This is not a valid equation; omit.
Prime factorization: The expression of an integer greater than one as a product of primes, unique up to order. The conjecture asks whether the prime factorizations of all three bases overlap.
Pillai's conjecture: The conjecture that the difference between consecutive perfect powers tends to infinity, with only finitely many solutions for each fixed difference. It concerns another scarcity question for equations involving distinct perfect powers.
Greatest common divisor: The largest positive integer that divides each of a given set of integers. A shared prime factor is equivalent to the bases having a greatest common divisor greater than one.
1^m: The number one raised to any integer power is one. Bases equal to one can produce equations without a shared prime factor, so formulations must account for them.