# Questions tagged [prime-numbers]

Questions where prime numbers play a key-role, such as: questions on the distribution of prime numbers (twin primes, gaps between primes, Hardy–Littlewood conjectures, etc); questions on prime numbers with special properties (Wieferich prime, Wolstenholme prime, etc.). This tag is often used as a specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.

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### On distribution of prime pairs coming from certain polynomials

Consider the polynomials $$g(x)=(2x)^4+((2x)^2+1)^2$$ $$h(x)=(2x)^4+((2x)^2-1)^2.$$ If $k$ odd integers $x_1,\dots,x_k$ are uniformly randomly chosen in $(t,2t)$ and the polynomials are evaluated at ...
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### How to explain this number-theoretic seeming “almost coincidence”?

For natural numbers $n\geq2$, let $d(n)$ be the number of divisors of $n$, and let \begin{equation} g(n)=n\sum_i r_i(p_i-1) \end{equation} where $n=\prod_i p_i^{r_i}$ is the factorisation of $n$ as a ...
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### Primes of the form power of 2 plus a prime

By Bertrand' postulate, there exists a prime between $2^n$ and $2^{n+1}$. For every $n$, is there a prime $p < 2^n$ such that $2^n+p$ is a prime? The smallest such primes are listed in OEIS A056206....
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### Calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$ [closed]

How to calculate the great common factor between $2^{2n+1}-1$ and $2^{4m+2}+1$, where $n$ and $m$ are positive numbers. We guess that: the great common factor is $1$.
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### Bins-and-primes (prime divisors of $\prod(a_i-a_j)$, II)

This is a refined version of a question I have recently posted. For a prime $p$, let $\varphi_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the ...
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### Prime divisors of $\prod(a_i-a_j)$

For a prime $p$, let $\varphi_p\colon\mathbb Z\to\mathbb Z/p\mathbb Z$ denote the canonical homomorphism from the integers onto the group of order $p$. Given an integer $n\ge 3$, what is the smallest ...
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### Convergence of Farey series integral of a "density" function as the order tends to infinity

Let $F_n$ denote the $n$-th Farey sequence, and let $q$ be a rational number such that $0 \leq q \leq 1$. I am studying the convergence of a specific integral related to Farey series, defined as ...
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Let $f:\mathbb N \to \mathbb C$ be an arithmetic function. There are various ways to define what the Siegel–Walfisz (S–W) property is for $f(n)$. One simple way is that there exists some function $g(... 0 votes 0 answers 29 views ### '$r$-prime factor number' distribution centering an '$l$-prime factor number' Let '$k$-prime factor integer'$n$be an integer of form $$2^t\prod_{i=1}^{k'} p_i^{e_i}$$ where$1\leq k'\leq k$and at every$i\in\{1,\dots,k\}p_i$is a distinct odd prime and$e_i\in\mathbb Z_{&...
The series $$\sum_{n=1}^\infty\frac{z^{p_n-n}}{n!}$$ associated to the sequence $p_1=2,p_2=3,p_3=5,p_4=7,p_5=11,\ldots$ of prime numbers defines a holomorphic function in the open disc of radius $e$. ...
What is the conjectured distributional behavior of semiprimes $pq$ ($p$ and $q$ are primes) having the property $2pq+1$ and $2pq-1$ are primes?