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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.09579 |
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| _version_ | 1866913807257829376 |
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| author | Jia, Zhengkun Li, Huixi Liu, Yushuo |
| author_facet | Jia, Zhengkun Li, Huixi Liu, Yushuo |
| contents | Erdős and Graham posed the question of whether there exists an integer $n$ such that the divisors of $n$ greater than $1$ form a distinct covering system with pairwise coprime moduli for overlapping congruences. Adenwalla recently proved no such $n$ exists, introducing the concept of nice integers, those where such a system exists without necessarily covering all integers. Moreover, Adenwalla established a necessary condition for nice integers: if $n$ is nice and $p$ is its smallest prime divisor, then $n/p$ must have fewer than $p$ distinct prime factors. Adenwalla conjectured this condition is also sufficient. In this paper, we resolve this conjecture affirmatively by developing a novel constructive framework for residue assignments. Utilizing a hierarchical application of the Chinese Remainder Theorem, we demonstrate that every integer satisfying the condition indeed admits a good set of congruences. Our result completes the characterization of nice integers, resolving an interesting open problem in combinatorial number theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09579 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Resolving Adenwalla's conjecture related to a question of Erdős and Graham about covering systems Jia, Zhengkun Li, Huixi Liu, Yushuo Number Theory 11B25(Primary)11B05, 11B30(Secondary ) Erdős and Graham posed the question of whether there exists an integer $n$ such that the divisors of $n$ greater than $1$ form a distinct covering system with pairwise coprime moduli for overlapping congruences. Adenwalla recently proved no such $n$ exists, introducing the concept of nice integers, those where such a system exists without necessarily covering all integers. Moreover, Adenwalla established a necessary condition for nice integers: if $n$ is nice and $p$ is its smallest prime divisor, then $n/p$ must have fewer than $p$ distinct prime factors. Adenwalla conjectured this condition is also sufficient. In this paper, we resolve this conjecture affirmatively by developing a novel constructive framework for residue assignments. Utilizing a hierarchical application of the Chinese Remainder Theorem, we demonstrate that every integer satisfying the condition indeed admits a good set of congruences. Our result completes the characterization of nice integers, resolving an interesting open problem in combinatorial number theory. |
| title | Resolving Adenwalla's conjecture related to a question of Erdős and Graham about covering systems |
| topic | Number Theory 11B25(Primary)11B05, 11B30(Secondary ) |
| url | https://arxiv.org/abs/2504.09579 |