Are the Collatz and abc conjectures related?
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866915564539084800 |
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| author | Rozier, Olivier |
| author_facet | Rozier, Olivier |
| contents | The Collatz and $abc$ conjectures, both well known and thoroughly studied, appear to be largely unrelated at first sight. We show that assuming the $abc$ conjecture true is helpful to improve the lower bound of integers initiating a particular type of Collatz sequences, namely finite sequences of a given length where all terms but one are odd with the usual ``shortcut'' form. To obtain sharper bounds in this context, we are led to consider a small subset of the $abc$-hits. Then, it turns out that Collatz iterations as well as Wieferich primes may be used to find large triples in this subset. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_15284 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Are the Collatz and abc conjectures related? Rozier, Olivier Number Theory The Collatz and $abc$ conjectures, both well known and thoroughly studied, appear to be largely unrelated at first sight. We show that assuming the $abc$ conjecture true is helpful to improve the lower bound of integers initiating a particular type of Collatz sequences, namely finite sequences of a given length where all terms but one are odd with the usual ``shortcut'' form. To obtain sharper bounds in this context, we are led to consider a small subset of the $abc$-hits. Then, it turns out that Collatz iterations as well as Wieferich primes may be used to find large triples in this subset. |
| title | Are the Collatz and abc conjectures related? |
| topic | Number Theory |
| url | https://arxiv.org/abs/2306.15284 |