Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.17601 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916914782011392 |
|---|---|
| author | Le, Maohua Miyazaki, Takafumi |
| author_facet | Le, Maohua Miyazaki, Takafumi |
| contents | We study purely exponential Diophantine equations with four terms of consecutive bases. Notably, we prove that all solutions to the equation \[ n^x=(n+1)^y+(n+2)^z+(n+3)^w \] in positive integers $n,x,y,z$ and $w$ are given by $(n,x,y,z,w)=(2,5,1,1,2)$, $(3,3,2,1,1)$. Our proof of this result for each $n \ge 4$ provides an explicit modulus $M$ such that the corresponding equation has no solution already modulo $M$. This contributes to a classical problem posed by T. Skolem in 1930's on a local-global principle on purely exponential Diophantine equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17601 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Purely exponential Diophantine equations with four terms of consecutive bases: contribution to Skolem's conjecture Le, Maohua Miyazaki, Takafumi Number Theory 11D61 We study purely exponential Diophantine equations with four terms of consecutive bases. Notably, we prove that all solutions to the equation \[ n^x=(n+1)^y+(n+2)^z+(n+3)^w \] in positive integers $n,x,y,z$ and $w$ are given by $(n,x,y,z,w)=(2,5,1,1,2)$, $(3,3,2,1,1)$. Our proof of this result for each $n \ge 4$ provides an explicit modulus $M$ such that the corresponding equation has no solution already modulo $M$. This contributes to a classical problem posed by T. Skolem in 1930's on a local-global principle on purely exponential Diophantine equations. |
| title | Purely exponential Diophantine equations with four terms of consecutive bases: contribution to Skolem's conjecture |
| topic | Number Theory 11D61 |
| url | https://arxiv.org/abs/2508.17601 |