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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46
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3k views

The topology of Arithmetic Progressions of primes

The primary motivation for this question is the following: I would like to extract some topological statistics which capture how arithmetic progressions of prime numbers "fit together" in a manner ...
21
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1k views

Permutations of $(Z/pZ)^*$

Let $p$ be a prime integer, and let $(\mathbb Z/p\mathbb Z)^*$ be the set of non-zero elements of $\mathbb Z/p \mathbb Z$. Denote by $S((\mathbb Z/p \mathbb Z)^*)$ the group of permutations of ...
14
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0answers
841 views

Small primes attract large primes

I posted a version of this to stackexchange and got 12 up-votes and no answers in somewhat more than a day. Someone in a comment construed it as asking for a lot of novel research including figuring ...
9
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0answers
476 views

Are the twin primes the only positive double zeros of this real function?

Agno's answer was extremely helpful. For $x \in \mathbb{R}, x \ge 1$ define $$ f(x) = \sin\frac{\pi(\Gamma(x)+1)}{\lfloor x \rfloor}$$ By Wilson's theorem the positive integer zeros of $f(x)$ are ...
8
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0answers
119 views

Does the given operation on pairs of primes always repeat?

Let $p$ and $q$ be two distinct primes. The set $$A(p,q) =\{m+n : mp+nq=1 \textrm{ and } m,n \in \mathbb{Z}\}$$ is an arithmetic progression. Its step size $p-q$ is coprime to a fixed $m+n$ because ...
8
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339 views

Sieve bound for prime $k$-tuples

Let $d_1<d_2<\dots<d_k$ be integers. Then the number of integers $n\leq x$, such that $n+d_1, n+d_2, \ldots, n+d_k$ are simultaneously prime, is bounded above by $$ \mathfrak{S}(d_1, \ldots, ...
8
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0answers
664 views

Would the following conjectures imply $\lim\inf_{n\to\infty}p_{n+k}-p_{n}=O(k\log k)$?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, ...
7
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157 views

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 ...
7
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0answers
272 views

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 ...
7
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0answers
512 views

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 ...
6
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206 views

Does the Maynard-Tao Theorem apply to general tuples of linear forms?

In the paper http://arxiv.org/pdf/1311.5319v1.pdf the author states the following theorem, which he attributes to Maynard and Tao. For any integer $m > 2$, there exists an integer $k = k(m)$ such ...
6
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0answers
278 views

Twin Primes that are Sophie Germain Primes

Suppose $p$ is a prime such that $p + 2$ is also prime, and nothing else is known about $p$. Is there any reason to think that this affects the probability that $p$ is also a Sophie Germain prime? ...
6
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228 views

Generalizing prime numbers to product-indecomposable objects in toposes

Any object $A$ in any topos decomposes as $A\times1$ and $1\times A$, and in $FinSet$ objects with no other product decompositions are "prime numbers". Is there any extension of the theory of ...
6
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596 views

Would the following conjectures imply Cramer's conjecture?

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\inf\{r\le n-2, ...
6
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0answers
464 views

Any published references for this $O(n)$ time, $O(n^\epsilon)$ space identity for the count of primes?

I would like to know if this identity (or trivial equivalents) for $\pi(n)$, the count of primes, is currently published anywhere. $D_{0,a}(n) = 1$ $D_{1,a}(n) = \lfloor n\rfloor-a-1 \ \ \ \ \ \ \ \ ...
6
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0answers
341 views

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 ...
6
votes
0answers
590 views

“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 ...
5
votes
0answers
213 views

On the sum of consecutive primes and product of first and last

Lets consider the sequence of consecutive prime numbers $p_1=2 , p_2=3 ,p_4=5 , \cdots$ . $(p_n,p_{j})$ is to be called good prime pair if $$\sum_{i =n }^{j}p_i= p_n p_{j}$$ Meaning the sum of set of ...
5
votes
0answers
229 views

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 ...
5
votes
0answers
238 views

Should I expect to see numbers this smooth?

I have a sequence $N_k$ of numbers whose growth I wish to determine, or at least approximate nicely. When I look at the ratios of consecutive members, I find some interesting simplifications ...
5
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0answers
241 views

$n\varphi(n)\equiv 2\pmod{\sigma(n)}$ 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 ...
5
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0answers
295 views

How many ways can a number be Fortunate?

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 ...
5
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0answers
243 views

Picking out an odd subset

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 ...
4
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0answers
153 views

Can the following quantitative version of Chen's theorem be obtained?

The theorem of the Chinese mathematician Chen Jingrun is currently one of the best results with respect to the binary Goldbach problem. It asserts that every even integer $n$ is in the sumset $P + ...
4
votes
0answers
300 views

About sign changes of Li(x)-π(x)

Given a constant $C$, which are the best known upper bounds for the number of sign changes of the function $$ f: \mathbb{N} \rightarrow \mathbb{R}, \ \ x \mapsto {\rm Li}(x)-\pi(x) $$ in the range ...
4
votes
0answers
81 views

On a weighted sum in a lemma for sieve methods

I'm reading James Maynard's paper "Small gaps between primes". Lemma 6.1 (p.14) in this paper confused me. This lemma was taken from Goldston-Graham-Pintz-Yildirim's paper "Small gaps between ...
4
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0answers
208 views

Relative Densities

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count ...
4
votes
0answers
317 views

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 ...
4
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0answers
307 views

Estimate on radical of $2^n \pm 1$

Not sure if this belongs to MO or not. Are there any lower bound on radical of $2^n \pm 1$ (Recall that radical of integer $rad(k)$ is a product of primes which divide integer $k$)? As an example If ...
4
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0answers
145 views

smallest k such that highest prime factor of m(m+1)…(m+k-1) is > n if m > n.

I am fascinated by this entry OEIS A213253 which lists the smallest $k$ such that highest prime factor of $m(m+1)\dots(m+k-1)$ is $> n$ if $m > n$. The article has references to the algorithm ...
4
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0answers
327 views

Recursivity of the primes

As is well-known, with the finite set of the first $n$ primes $p_1,\dots ,p_n$ one can find all primes exactly (i.e. without false positives) with the Eratosthenes Sieve in the interval ...
4
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0answers
283 views

Generalization of Tamarkin’s ARO 1993, final round, problem 10/8: part II

Let us use the notations of my previous question about Tamarkin's problem. Let $\ell\in\left\lbrace 0,1,...,p\right\rbrace$. An element $f\in \mathbb Z^{\mathbb Z}$ is said to be ...
4
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0answers
354 views

Number of $k$-partitions of $n$ into odd prime parts

Browsing through OESIS I have found that if $p_p(n)$ denotes the number of partitions of $n$ into prime parts then $p_p(n) = O(e^{\frac{2 \Pi}{\sqrt{3}}\sqrt{n/\log n}})$. I am interested in the ...
4
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0answers
503 views

Are there an infinite number of prime Euclid numbers?

A number defined as the product of first $n$ prime numbers $+1$ is called $n$th Euclid number. Are there any survey on the progress for answering the following question: are there an infinite number ...
3
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0answers
87 views

Numbers expressible as sums of prime powers larger than n

Given a fixed $n \in \mathbb{N}$ larger than $1$, let $G(n)$ denote the largest number that is not expressible as a sum of prime powers larger than $n$ (the 'base' prime of the prime power need not be ...
3
votes
0answers
179 views

Logarithms of ratios of squarefree numbers

Let $M \geq 1$, and $N=2^M$. Let $a_1, \dots, a_N$ be the set of all the numbers that you get when forming all square-free products of the first $M$ primes. For example, for $M=2$ and $N=4$ you get ...
3
votes
0answers
189 views

Generating function for the characteristic function of prime numbers

What do we know about the generating function of $\chi(n)$ (A010051) $$ f(x) = \sum_{n=0}^\infty \chi(n)x^n = \sum_{p\text{ prime}} x^p $$ for $\chi(n)$ the characteristic function of the primes: ...
3
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0answers
138 views

Number of Inverse Pairs Modulo Prime $p$

Is there a result which gives a lower bound on the number of inverse pairs $(a, a^{-1})$ modulo prime $p$ lying in the interval $[1,t]$, where $t < p$?
3
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0answers
310 views

Linear independence over Q of logarithmic powers of prime numbers

I denote $p_k$ the $k^{th}$ prime number ($p_1=2$, etc...) Clearly, for any $n\in \mathbb{N}^*$, $(\log p_k)_{1\leq k\leq n}$ is linearly independent over $\mathbb{Q}$. My question concerns a ...
3
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0answers
489 views

Paul Erdős and Ramanujan Primes

It's easy to find Ramanujan's proof of Ramanujan primes: Ramanujan's Proof Wikipedia mentions that Paul Erdős also had a proof: Wikipedia article on Bertrand's Postulate Does anyone know the ...
3
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0answers
547 views

Least Prime Factor in a sequence of 2n consecutive integers

I was thinking about consecutive integers and I wondered if anyone had done work exploring whether a sequence of $2n$ consecutive integers (i.e. 101,102,103,...,100+2n) always contains at least one ...
2
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0answers
140 views

Arguments for the second Hardy–Littlewood conjecture being false?

Assume that $x,y > 2$, and that $x<y$. Then the second Hardy–Littlewood conjecture states that $$\pi(x + y) - \pi(y) \leq \pi(x).$$ We can easily justify this heuristically, since $$ ...
2
votes
0answers
81 views

Primality Criterion for Specific Class of Numbers of the Form kb^n-1

Let $N=k\cdot b^n-1$ where $b$ is an even integer , $3\nmid b$ , $3\nmid N$ , $k \equiv 1,5 \pmod{6}$ , $k< b^n $ and $n>2$ . Let $S_i=P_b(S_{i-1})$ with $S_0=P_{k\cdot b/2}(P_{b/2}(4))$ , ...
2
votes
0answers
86 views

Primality Criterion for Specific Classes of Generalized Fermat Numbers

Let $F_n(b)= b^{2^n}+1 $ where $b$ is an even integer , $ 3\nmid b , 5\nmid b $ and $n\ge2$ Let $S_i=P_b(S_{i-1})$ with $S_0=P_{b/2}(P_{b/2}(8))$ where ...
2
votes
0answers
338 views

New proofs of Euclid's theorem of the infinitude of primes?

Playing around with elementary inclusion-exclusion, I arrived at two simple variations of proofs of Euclid's theorem that I thought would be long known in the literature. So far I haven't been able to ...
2
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0answers
211 views

Relation between Maier's theorem and a conjecture of Montgomery and Soundararajan

Let us consider the number of primes in the interval $[N,N+h]$, with $h\leq N$. According to the answer given by Lucia to a previous question on the distribution of primes, it is natural to consider ...
2
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0answers
122 views

n-ary quadratic forms with $S$-integer values

Let $Q(x_1,\ldots,x_n):=x_1^2+\cdots+x_n^2$ be an $n$-ary quadratic form. Given a finite set of (rational) primes $S$ is there an algorithm or theorem that describes all solutions to ...
2
votes
0answers
117 views

Analytic varieties for the primes and the twin primes

I am wondering what real and complex analysis say about the primes and twin primes. According to Wikipedia analytic variety is defined locally as the set of common zeros of finitely many analytic ...
2
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0answers
262 views

A question concerning the strange arithmetic derivation

This question is related to Strange (or stupid) arithmetic derivation. The original question whether an unbounded sequence of iterates exists is still unanswered. $$n=\prod_{i=1}^{k}p_i^{\alpha_i} ...
2
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0answers
195 views

Efficient ways to count primes satisfying Zhang's theorem

The theorem of Yitang Zhang states that there exist a finite $k \in \mathbb{N}$ such that there exist infinitely pairs of primes $(p,q)$ such that $|p - q| \leq k$. The statement that $k$ can be taken ...