On Characterizing Potential Friends of 20
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909787339358208 |
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| author | Chatterjee, Tapas Mandal, Sagar Mandal, Sourav |
| author_facet | Chatterjee, Tapas Mandal, Sagar Mandal, Sourav |
| contents | Does $20$ have a friend? Or is it a solitary number? A folklore conjecture asserts that $20$ has no friends i.e. it is a solitary number. In this article, we prove that, a friend $N$ of $20$ is of the form $N=2\cdot5^{2a}\cdot m^2$, with $(3,m)=(7,m)=1$ and it has at least six distinct prime divisors. Furthermore, we show that $Ω(N)\geq 2ω(N)+6a-5$ and if $Ω(m)\leq K$ then $N< 10\cdot 6^{(2^{K-2a+3}-1)^2}$, where $Ω(n)$ and $ω(n)$ denote the total number of prime divisors and the number of distinct prime divisors of the integer $n$ respectively. In addition, we deduce that, not all exponents of odd prime divisors of friend $N$ of $20$ are congruent to $-1$ modulo $f$, where $f$ is the order of $5$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ such that $3\mid f$ and $p$ is a prime congruent to $1$ modulo $6$. Also, we prove necessary upper bounds for all prime divisors of friends of 20 in terms of the number of divisors of the friend. In addition, we prove that, if $P$ is the largest prime divisor of $N$ then $P<N^{\frac{1}{4}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04451 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Characterizing Potential Friends of 20 Chatterjee, Tapas Mandal, Sagar Mandal, Sourav General Mathematics 11A25 Does $20$ have a friend? Or is it a solitary number? A folklore conjecture asserts that $20$ has no friends i.e. it is a solitary number. In this article, we prove that, a friend $N$ of $20$ is of the form $N=2\cdot5^{2a}\cdot m^2$, with $(3,m)=(7,m)=1$ and it has at least six distinct prime divisors. Furthermore, we show that $Ω(N)\geq 2ω(N)+6a-5$ and if $Ω(m)\leq K$ then $N< 10\cdot 6^{(2^{K-2a+3}-1)^2}$, where $Ω(n)$ and $ω(n)$ denote the total number of prime divisors and the number of distinct prime divisors of the integer $n$ respectively. In addition, we deduce that, not all exponents of odd prime divisors of friend $N$ of $20$ are congruent to $-1$ modulo $f$, where $f$ is the order of $5$ in $(\mathbb{Z}/p\mathbb{Z})^\times$ such that $3\mid f$ and $p$ is a prime congruent to $1$ modulo $6$. Also, we prove necessary upper bounds for all prime divisors of friends of 20 in terms of the number of divisors of the friend. In addition, we prove that, if $P$ is the largest prime divisor of $N$ then $P<N^{\frac{1}{4}}$. |
| title | On Characterizing Potential Friends of 20 |
| topic | General Mathematics 11A25 |
| url | https://arxiv.org/abs/2409.04451 |