# Tagged Questions

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 ...

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### Does $\binom{2n}{n} \equiv 2 \pmod p$ ever hold?

Well, the title does not tell the whole story; the complete question is: Are there any primes of the form $p=2n(n-1)+1$, with integer $n\ge 1$, such that $$\binom{2n}{n} \equiv 2\pmod p ?$$ ...
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### Independence between the number of prime factors of $n$ and $n+2$

I am interested in having an upper bound for the cardinality of $\#\left\{n\leq x\,:\quad\omega(n)=k, \omega(n+2)=\ell\right\}$ for $k,\ell\geq 1$, where $\omega(n)=\sum_{p\vert n}1$ counts the ...
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### Two different ways to count Mersenne Primes

Hi there, the motivation for this question is to better understand the heuristics of Mersenne primes, and I was motivated by the recent questions (Mersenne quasi-primes) and (Primes in generalized ...
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### k-Almost Primes in short intervals

According to this question every interval $[x, x + x^{0.45}]$ contains a product of two primes, and this has been improved further slightly. Are there better results available for $k$-almost primes? ...
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### Can primes be (almost) random sequence in von Mises sense?

Random models for primes (such as Cramer's model) have been extensively used for informal justification of various conjectures involving primes. It is crucial to understand in what sense sequence of ...
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### Are there an infinite number of twin semiprimes?

A semiprime is a number that is the product of two (possibly equal) primes. Define twin semiprimes (my terminology) as two consecutive numbers both semiprimes. For example, $(57,58)$ are twin ...
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### “Forthcoming paper” of Goldston-Graham-Pintz-Yıldırım

The above-named authors of [1] and its (significantly different) published version [2] write: In a forthcoming paper, we will show how the methods here can be extended to prove corresponding ...
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### On the ratio of Gilbreath sequences

Definitions: let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. ...
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### In which orders can the numbers of prime factors of consecutive integers be?

Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur. Given ...
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### Montgomery's conjecture and lower bound on certain Fourier transform.

Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...
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### Primes of the form $x^2+ny^2$ such that when you swap x and y you get another prime

Hello, Im looking at primes of the form $x^2+ny^2$ for $n>1$ where we can swap $x$ and $y$ and get another prime, I have found many such pairs for many values of n, and i wanted to know if there ...
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### “probabilistic” density of primes?

A certain set $\cal P$ of primes is defined by two assumedly independent conditions: The first condition on a prime $p$ can be characterized in terms of the type of splitting of $p$ in certain Galois ...
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### Primes for which 2 is a primitive root

I am writing a paper in which I keep referring to primes p for which 2 is a primitive root mod p and so I want to give a name for these primes. Is there a name for these primes in the literature ...
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### $n\varphi(n)\equiv 2\pmod{\sigma(n)}$ as a primality test

It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of ...
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### AKS Algorithm Pseudoprimes

The AKS algorithm is based on the following fully deterministic primality check: Let input $n>1$ and $a \in \mathbb{N}$ such that $(a,n)=1$. Then $n$ is prime if and only if \tag{1}(x+a)^n \...
An integer $m$ is Fortunate if it can be written as $q-P$, where $P$ is a primorial and $q$ is the smallest prime greater than $P+1$. It is conjectured that Fortunate numbers are always prime. It ...
I want to know whether the following property holds: There exists a constant C such that for any big positive integer N and any nonempty subset M of {1,...,N} there exists a positive integer p(M)$\lt$...