Exceptional set in the abc conjecture
We study solutions to the equation \(a + b = c\), where \(a, b, c\) form a triple of coprime natural numbers. The well-known abc conjecture asserts that, for any \(\varepsilon > 0\), such triples satisfy \(\text{rad}(abc) \geq c ^ {1−\varepsilon}\) with finitely many exceptions. In this ongoing project we aim to formalise a power-saving bound on the exceptional set of triples. Specifically, we show that there are \(O(X^{33/50})\) integer triples \((a, b, c) ∈ [1, X]^3\), which satisfy \(\text{rad}(abc) < c ^ {1−\varepsilon}\). The proof is based on a combination of bounds for the density of integer points on varieties, coming from the determinant method, Thue equations, geometry of numbers, and Fourier analysis. We will follow the paper of Browning, Lichtman and Teräväinen
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