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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2310.15900 |
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| _version_ | 1866911818674339840 |
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| author | Thackeray, Henry Robert |
| author_facet | Thackeray, Henry Robert |
| contents | For each positive integer n, if the sum of the factors of n is divided by n, then the result is called the abundancy index of n. If the abundancy index of some positive integer m equals the abundancy index of n but m is not equal to n, then m and n are called friends. A positive integer with no friends is called solitary. The smallest positive integer that is not known to have a friend and is not known to be solitary is 10.
It is not known if the number 6 has odd friends, that is, if odd perfect numbers exist. In a 2007 article, Nielsen proved that the number of nonidentical prime factors in any odd perfect number is at least 9. A 2015 article by Nielsen, which was more complicated and used a computer program that took months to complete, increased the lower bound from 9 to 10.
This work applies methods from Nielsen's 2007 article to show that each friend of 10 has at least 10 nonidentical prime factors.
This is a formal write-up of results presented at the Southern Africa Mathematical Sciences Association Conference 2023 at the University of Pretoria. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_15900 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Each friend of 10 has at least 10 nonidentical prime factors Thackeray, Henry Robert Number Theory 11A25 For each positive integer n, if the sum of the factors of n is divided by n, then the result is called the abundancy index of n. If the abundancy index of some positive integer m equals the abundancy index of n but m is not equal to n, then m and n are called friends. A positive integer with no friends is called solitary. The smallest positive integer that is not known to have a friend and is not known to be solitary is 10. It is not known if the number 6 has odd friends, that is, if odd perfect numbers exist. In a 2007 article, Nielsen proved that the number of nonidentical prime factors in any odd perfect number is at least 9. A 2015 article by Nielsen, which was more complicated and used a computer program that took months to complete, increased the lower bound from 9 to 10. This work applies methods from Nielsen's 2007 article to show that each friend of 10 has at least 10 nonidentical prime factors. This is a formal write-up of results presented at the Southern Africa Mathematical Sciences Association Conference 2023 at the University of Pretoria. |
| title | Each friend of 10 has at least 10 nonidentical prime factors |
| topic | Number Theory 11A25 |
| url | https://arxiv.org/abs/2310.15900 |