# Questions tagged [probabilistic-number-theory]

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### In the modular exponentiation, as used with the adaptive root problem, how to chose the best base that will yield as few results as possible?

Let $n,m,w\in\Bbb N$ and $\lambda\in\Bbb P$ such that $w^\lambda \mod m = n$, with the requirements: $\lambda$ being a random large prime such as $w^\lambda > 2\times m$ $1 < n < m−1$. m is ...
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### Integers with $k$ prime factors, in terms of the Möbius function

A $k$-free integer is an integer $n$ such that there is no $k$th power dividing $n$. It is well known (see Murty's Problems in Analytic Number Theory q1.18 for instance) that \sum_{d^k|...
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### Existence of an $\alpha$-regular measure with positive measure on a binary digits do not have a limiting frequency

let $$X=\left\{ \sum_{n=1}^{\infty}a_{n}2^{-n}:a_{n}\in\left\{ 0,1\right\} ,\liminf\frac{1}{n}\sum_{i=1}^{n}a_{i}<\limsup\frac{1}{n}\sum_{i=1}^{n}a_{i}\right\}$$ I'm studying fractal geometry and ...
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### Is there a Cramer's conjecture for Sophie Germain primes?

A prime $q$ such that $2q+1$ is also a prime is a Sophie Germain prime. Cramer's conjecture tells gap between consecutive primes is bound by $O(\log^2p)$. Is there a similar conjecture for Sophie ...
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### $L^\infty$ norm lower bound for Integer points in null spaces of recursively defined integer vectors?

Letting $\otimes$ be matrix kronecker/tensor product with $n\in\Bbb N$ as a parameter define non-negative integer vectors recursively $$v_1=\begin{bmatrix}a_1&b_1\end{bmatrix}\in\Bbb Z_{>0}^2$$ ...
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Divisor function $d(n,m)$ counts the number of $q\in\Bbb N$ with $b<q<m$ such that $n\bmod q\equiv0$. Given $b>0$ what is the correct asymptotic, probabilistic and average case behavior of ...