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Given $P(x) \in \mathbb{Z}[x]$ a polynomial, $n$ a positive integer and $p$ a prime, there is a result that relates $|\text{Im } P \pmod{p^n}|$ with $|\text{Im } P \pmod{p^{n+1}}|$ perhaps in terms of ...

Let $+,\cdot$ denote multiplication in $\mathbb{N}_0$. The addition and multiplication in $\operatorname{On}_p$ are denoted $\oplus, \otimes$. Recursive definition of addition: $$x \oplus y := ((x+y) \...

I am interested in knowing about the distribution of exponentially high order elements in $\mathbb{F}_p$. To be precise let $s$ be of the order $\frac{p}{\log^{k}(p)}$ for some fixed $k$ and integer. ...

The collection of prime powers can be characterized in the following way: There is a field with $q$ elements if and only if $q$ is a prime power. Furthermore if it exists then this field is unique ...