On the Sum of Squarefree Integers and a Power of Two
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917827825369088 |
|---|---|
| author | Hercher, Christian |
| author_facet | Hercher, Christian |
| contents | Erdos conjectured that every odd number greater than one can be expressed as the sum of a squarefree number and a power of two. Subsequently, Odlyzko and McCranie provided numerical verification of this conjecture up to $10^7$ and $1.4\cdot 10^9$. In this paper, we extend the verification to all odd integers up to $2^{50}$, thereby improving the previous bound by a factor of more than $8\cdot 10^5$. Our approach employs a highly parallelized algorithm implemented on a GPU, which significantly accelerates the process. We provide details of the algorithm and present novel heuristic computations and numerical findings, including the smallest odd numbers $<2^{50}$ that require a higher power of two as all smaller ones in their representation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_01964 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Sum of Squarefree Integers and a Power of Two Hercher, Christian Number Theory 11A67, 11Y99 Erdos conjectured that every odd number greater than one can be expressed as the sum of a squarefree number and a power of two. Subsequently, Odlyzko and McCranie provided numerical verification of this conjecture up to $10^7$ and $1.4\cdot 10^9$. In this paper, we extend the verification to all odd integers up to $2^{50}$, thereby improving the previous bound by a factor of more than $8\cdot 10^5$. Our approach employs a highly parallelized algorithm implemented on a GPU, which significantly accelerates the process. We provide details of the algorithm and present novel heuristic computations and numerical findings, including the smallest odd numbers $<2^{50}$ that require a higher power of two as all smaller ones in their representation. |
| title | On the Sum of Squarefree Integers and a Power of Two |
| topic | Number Theory 11A67, 11Y99 |
| url | https://arxiv.org/abs/2411.01964 |