Bounds on the exceptional set in the $abc$ conjecture
| Autor: | Browning, Tim, Lichtman, Jared Duker, Teräväinen, Joni |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We study solutions to the equation $a+b=c$, where $a,b,c$ form a triple of coprime natural numbers. The $abc$ conjecture asserts that, for any $\epsilon>0$, such triples satisfy $\mathrm{rad}(abc) \ge c^{1-\epsilon}$ with finitely many exceptions. In this article we obtain a power-saving bound on the exceptional set of triples. Specifically, we show that there are $O(X^{33/50})$ integer triples $(a,b,c)\in [1,X]^3$, which satisfy $\mathrm{rad}(abc) < c^{1-\epsilon}$. 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. Comment: 17 pages |
| Databáze: | arXiv |
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