The $abc$ conjecture is true almost always
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912383078760448 |
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| author | Lichtman, Jared Duker |
| author_facet | Lichtman, Jared Duker |
| contents | Let ${\rm rad}(n)$ denote the product of distinct prime factors of an integer $n\geq 1$. The celebrated $abc$ conjecture asks whether every solution to the equation $a+b=c$ in triples of coprime integers $(a,b,c)$ must satisfy ${\rm rad}(abc) > K_\varepsilon\, c^{1-\varepsilon}$, for some constant $K_\varepsilon>0$. In this expository note, we present a classical estimate of de Bruijn that implies almost all such triples satisfy the $abc$ conjecture, in a precise quantitative sense. Namely, there are at most $O(N^{2/3})$ many triples of coprime integers in a cube $(a,b,c)\in\{1,\ldots,N\}^3$ satisfying $a+b=c$ and ${\rm rad}(abc) < c^{1-\varepsilon}$. The proof is elementary and essentially self-contained. Beyond revisiting a classical argument for its own sake, this exposition is aimed to contextualize a new result of Browning, Lichtman, and Teräväinen, who prove a refined estimate $O(N^{33/50})$, giving the first power-savings since 1962. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_13991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The $abc$ conjecture is true almost always Lichtman, Jared Duker Number Theory Algebraic Geometry Combinatorics 11D45 (11D41, 11D75) Let ${\rm rad}(n)$ denote the product of distinct prime factors of an integer $n\geq 1$. The celebrated $abc$ conjecture asks whether every solution to the equation $a+b=c$ in triples of coprime integers $(a,b,c)$ must satisfy ${\rm rad}(abc) > K_\varepsilon\, c^{1-\varepsilon}$, for some constant $K_\varepsilon>0$. In this expository note, we present a classical estimate of de Bruijn that implies almost all such triples satisfy the $abc$ conjecture, in a precise quantitative sense. Namely, there are at most $O(N^{2/3})$ many triples of coprime integers in a cube $(a,b,c)\in\{1,\ldots,N\}^3$ satisfying $a+b=c$ and ${\rm rad}(abc) < c^{1-\varepsilon}$. The proof is elementary and essentially self-contained. Beyond revisiting a classical argument for its own sake, this exposition is aimed to contextualize a new result of Browning, Lichtman, and Teräväinen, who prove a refined estimate $O(N^{33/50})$, giving the first power-savings since 1962. |
| title | The $abc$ conjecture is true almost always |
| topic | Number Theory Algebraic Geometry Combinatorics 11D45 (11D41, 11D75) |
| url | https://arxiv.org/abs/2505.13991 |