All Questions

For odd integer $n$ define the function $$ J(n)=(2^{\varphi(n)}-1) \bmod{n^2}$$ $J(n)$ is integer in $[0,n^2-1]$ and it is divisible by $n$. Integer $n$ is Wieferich number iff $J(n)=0$ and if $n$ is ...

For natural $n$, define the sequence $$ a(n)=\gcd(2^n-1,\phi(2^n-1)) $$ It doesn't appear to be in OEIS and starts $1,1,1,1,9,1,1,1,3,1,9,1,3,1,1,1,27,1,75,49$ Q1 Can we unconditionally prove $a(n)=1$...

Factorization is not known to have a polynomial time algorithm. Traditionally the input length is number of bits in representation of the integer to be factored. However now consider integers of form $...

There are many results about consecutive integers all having small prime factors. But what about consecutive integers each of which has a large prime factor? More precisely, let $P(n)$ be the ...

I apologize if this is a naive question about greatest prime factors (gpf). I was thinking about the sequence of integers where $\mathrm{gpf}(x) \le p$ where $p$ is any prime. Clearly, as $x$ ...