For every real number epsilon > 0, there should be a constant
K(epsilon) such that every triple of pairwise coprime positive integers
a, b, c satisfying a + b = c obeys
c < K(epsilon) * rad(abc)^(1 + epsilon).
Here rad(n) is the product of the distinct primes dividing n.
The radical forgets prime multiplicities: rad(72) = rad(2^3*3^2) = 6.
The conjecture says that a sum a+b=c cannot usually produce a number with
very large repeated prime powers compared with the set of primes involved.
The conjecture has powerful consequences for Diophantine equations, including effective forms of results related to Fermat-type equations, integral points, and bounds on exceptional arithmetic behaviour.
There are strong heuristic and conditional results, and a claimed proof using inter-universal Teichmueller theory has generated extensive discussion. This repository treats the conjecture as an open verification problem unless a complete proof is independently understood and accepted by the reader.
The statement quantifies over all coprime triples and permits an arbitrary
epsilon. Elementary factorisation sees individual triples but gives no
uniform control over their prime-power structure.
- Compute
rad(abc)for small coprime examples. - Prove elementary consequences of unique factorisation used in the setup.
- Study the link between abc and the arithmetic of elliptic curves.
- Distinguish a numerical search for exceptions from a proof of the uniform inequality.
The expression log(c)/log(rad(abc)) measures the quality of a triple. The
conjecture permits occasional unusually good triples, but says their quality
cannot stay above 1+epsilon without a constant depending on epsilon.
The coprimality hypothesis is essential: common factors can otherwise be
removed before measuring the radical.
For any proposed consequence, identify exactly where the epsilon and its
constant enter. Check that the argument does not silently replace “for every
epsilon” by a fixed exponent-one bound. Verify all gcd reductions and separate
effective statements from ineffective existence statements.
Start with an elementary treatment of radicals and height functions, then the Mason--Stothers theorem for polynomials, often called the function-field abc theorem. It is an illuminating analogue but is not a proof of the integer case.