An effective estimate for the sum of two cubes problem
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909220103782400 |
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| author | Bhattacharjee, Saunak |
| author_facet | Bhattacharjee, Saunak |
| contents | Let $f(x, y) \in \mathbb{Z}[x, y]$ be a cubic form with non-zero discriminant, and for each integer $m \in \mathbb{Z}$, let, $N_{f}(m)=\#\left\{(x, y) \in \mathbb{Z}^{2}: f(x, y)=m\right\} $. In 1983, Silverman proved that $N_{f}(m)>Ω\left((\log |m|)^{3 / 5}\right)$ when $f(x, y)=x^{3}+y^{3}$. In this paper, we obtain an explicit bound for $N_f(m)$, namely, showing that $N_{f}(m)>4.2\times 10^{-6}(\log |m|)^{11/13}$ (holds for infinitely many integers m), when $f(x, y)=x^{3}+y^{3}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17955 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An effective estimate for the sum of two cubes problem Bhattacharjee, Saunak Number Theory 11G50 Let $f(x, y) \in \mathbb{Z}[x, y]$ be a cubic form with non-zero discriminant, and for each integer $m \in \mathbb{Z}$, let, $N_{f}(m)=\#\left\{(x, y) \in \mathbb{Z}^{2}: f(x, y)=m\right\} $. In 1983, Silverman proved that $N_{f}(m)>Ω\left((\log |m|)^{3 / 5}\right)$ when $f(x, y)=x^{3}+y^{3}$. In this paper, we obtain an explicit bound for $N_f(m)$, namely, showing that $N_{f}(m)>4.2\times 10^{-6}(\log |m|)^{11/13}$ (holds for infinitely many integers m), when $f(x, y)=x^{3}+y^{3}$. |
| title | An effective estimate for the sum of two cubes problem |
| topic | Number Theory 11G50 |
| url | https://arxiv.org/abs/2403.17955 |