论文标题
除数总和的算术特性
Arithmetic properties of the sum of divisors
论文作者
论文摘要
除数函数$σ（n）$表示正整数$ n $的除数的总和。对于Prime $ P $和$ M \ in \ Mathbb {n} $，$ P $ -M $的$ P $ ADIC估值是$ P $的最高功率，该功率将$ m $划分为$ m $。建立了$ν_{p}（σ（n））$的公式。对于$ p = 2 $，这些仅涉及划分$ n $的奇数。 These expressions are used to establish the bound $ν_{2}(σ(n)) \leq \lceil\log_{2}(n) \rceil$, with equality if and only if $n$ is the product of distinct Mersenne primes, and for an odd prime $p$, the bound is $ν_{p}(σ(n)) \leq \lceil \ log_ {p}（n）\ rceil $，具有与ljunggren-nagell diophantine方程解决方案有关的平等。
The divisor function $σ(n)$ denotes the sum of the divisors of the positive integer $n$. For a prime $p$ and $m \in \mathbb{N}$, the $p$-adic valuation of $m$ is the highest power of $p$ which divides $m$. Formulas for $ν_{p}(σ(n))$ are established. For $p=2$, these involve only the odd primes dividing $n$. These expressions are used to establish the bound $ν_{2}(σ(n)) \leq \lceil\log_{2}(n) \rceil$, with equality if and only if $n$ is the product of distinct Mersenne primes, and for an odd prime $p$, the bound is $ν_{p}(σ(n)) \leq \lceil \log_{p}(n) \rceil$, with equality related to solutions of the Ljunggren-Nagell diophantine equation.
