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| Format: | Preprint |
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2022
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| Online Access: | https://arxiv.org/abs/2207.14492 |
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| _version_ | 1866917967328968704 |
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| author | Gherga, Adela Siksek, Samir |
| author_facet | Gherga, Adela Siksek, Samir |
| contents | A Thue-Mahler equation is a Diophantine equation of the form $$F(X,Y) = a\cdot p_1^{z_1}\cdots p_v^{z_v}, \qquad \gcd(X,Y)=1$$ where $F$ be an irreducible homogeneous binary form of degree at least $3$ with integer coefficients, $a$ is a non-zero integer and $p_1, \dots, p_v$ are rational primes. Existing algorithms for resolving such equations require computations in the number field obtained by adjoining three roots of $F(X,1)=0$. We give a new algorithm that requires computations only in the number field obtained by adjoining one root, making it far more suited for higher degree examples. We also introduce a lattice sieving technique reminiscent of the Mordell--Weil sieve that makes it practical to tackle Thue--Mahler equations of higher degree and with larger sets of primes. We give several examples including one of degree $11$.
Let $P(m)$ denote the largest prime divisor of an integer $m \ge 2$. As an application of our algorithm we determine all pairs $(X,Y)$ of coprime non-negative integers such that $P(X^4-2Y^4) \le 100$, finding that there are precisely $49$ such pairs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_14492 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Efficient resolution of Thue-Mahler equations Gherga, Adela Siksek, Samir Number Theory 11D59 A Thue-Mahler equation is a Diophantine equation of the form $$F(X,Y) = a\cdot p_1^{z_1}\cdots p_v^{z_v}, \qquad \gcd(X,Y)=1$$ where $F$ be an irreducible homogeneous binary form of degree at least $3$ with integer coefficients, $a$ is a non-zero integer and $p_1, \dots, p_v$ are rational primes. Existing algorithms for resolving such equations require computations in the number field obtained by adjoining three roots of $F(X,1)=0$. We give a new algorithm that requires computations only in the number field obtained by adjoining one root, making it far more suited for higher degree examples. We also introduce a lattice sieving technique reminiscent of the Mordell--Weil sieve that makes it practical to tackle Thue--Mahler equations of higher degree and with larger sets of primes. We give several examples including one of degree $11$. Let $P(m)$ denote the largest prime divisor of an integer $m \ge 2$. As an application of our algorithm we determine all pairs $(X,Y)$ of coprime non-negative integers such that $P(X^4-2Y^4) \le 100$, finding that there are precisely $49$ such pairs. |
| title | Efficient resolution of Thue-Mahler equations |
| topic | Number Theory 11D59 |
| url | https://arxiv.org/abs/2207.14492 |