Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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Special Case of Composite mersenne number mod p

We want to investigate if a composite mersenne number $p|2^{qb}-1$ where $p\nmid2^q-1$ ,$q,p$ are primes, $p=1+6qb,\ qb\equiv1(mod64) $ and $b$ is an odd number. In general for $$\begin{align*} ...
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Test about prime gaps: which conclusions can be drawn from the results?

I did the following test: For every prime, take the prime gap distance $dp$ to the previous prime and the next prime $dn$, then calculate $a=(pp\ mod\ dp)$ and $b=(np\ mod\ dn)$. If $a$ or $b$ ...
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Let $f(x)$ be defined over all rationals $x$ in $[0,1]$ and let $F(n) = \sum_{i=1}^n f(\frac in)$

also define $$F^*(n) = \sum_{i=1\,\,(i,n)=1}^n f(\frac in)$$ then prove that $$F^* = \mu * F$$ where $\mu$ is the Möebius function and the $*$ means the Dirichlet convolution. I tried the Bell series ...