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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49
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4answers
4k views

Strange (or stupid) arithmetic derivation

Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...
11
votes
1answer
516 views

Are [Wieferich] primes the only solutions to the equation $2^{k-1} \equiv 1 \pmod{k^2}$?

While studying a certain Diophantine equation in the squarefree integer $k \ge 2$, I believe I have proven the necessary restriction $$2^{k-1} \equiv 1\!\!\pmod{k^2}. \qquad(\star)$$ Based on what ...
6
votes
1answer
454 views

On permuted sum of squares of primes in a list

We want to pick a set of distinct primes (if not possible, then just positive numbers) $p_1,p_2,\dots,p_k$ such that there exists $t$ permutations, ...
4
votes
0answers
220 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 ...
11
votes
0answers
624 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 ...
5
votes
2answers
321 views

Function with zeros plus/minus the primes

While playing with Cohen's pari script prodeulerrat found a function. For $s \in \mathbb{C}$ define $$ f(s) = \prod_{p \text{ prime}} (1-\frac{s^2}{p^2})$$ The product converges everywhere, no poles ...
4
votes
3answers
338 views

Set of primes dividing polynomials and composition

For a non-constant polynomial $A \in \mathbb{Z}[x]$, let $\mathcal{P}(A)$ denote the set of prime numbers $p$ which divide $A(n)$ for some integer $n$. If $\mathcal{P}(A) \subseteq \mathcal{P}(B)$ for ...
11
votes
2answers
2k views

An interaction between prime numbers

Let   $p_1\ p_2\ \ldots$ be the sequence of all natural prime numbers. There is a slight (just slight) but clear tendency for imitating the number of primes in an interval $(p_k;\ p_n)$   by ...
1
vote
0answers
307 views

Green-Tao style theorem for quadratic regressions (Ulam Spiral)

This is a naive question about number theory. Looking at an Ulam spiral which illustrates primes of the form e.g. $4x^2-2x+c$ and other quadratic equations $ax^2+bx+c$, with $c>0$, there appears a ...
5
votes
1answer
461 views

Other implications of Zhang's method

I have been reading a bit about Zhang's proof and the associated Polymath8 project. Though Tao's high level summary ...
0
votes
0answers
89 views

Sequences sharing some primitive prime divisors

Let $q=p^\alpha$ and $q'=p'^\alpha$. Moreover, define $r_i$ and $u_i$ as primitive prime divisor of $q^i-1$ and $q'^i-1$, respectively. Let $\{r_1\}=\{u_1\}$, $\{r_2\}=\{u_2\}$, $\{r_3\}=\{u_3\}$, ...
5
votes
3answers
702 views

a question for the prime counting function

A famous inequality that has been proved by J.B. Rosser and L. Schoenfeld says that $\frac{n}{\ln n-1/2}$ < $\pi(n)$<$\frac{n}{\ln n-3/2} , n\ge 67$. Using this inequality we can prove ...
1
vote
0answers
171 views

apparent contradiction from result on prime gaps [closed]

I'm looking at Theorem 3 of this paper, which is If $\:x\geq \exp(\exp(45))\:$ and $\;\;h\:\geq\:3\cdot x^{\frac23}\;\;$ then $$\pi(x+h)-\pi(x) \; \geq \; h\cdot \left(1-\left(3192.34\cdot ...
0
votes
0answers
111 views

Composite Euclid Numbers

Is there any way (general procedure, i mean) to determine if a Euclid Number (En = pn# + 1) is prime or composite? Any research papers exploring this theme are also welcome. Thanks!
8
votes
0answers
124 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
votes
1answer
317 views

How constructive is Dirichlet on primes in progressions?

Is there a known elementary function bound in terms of $a,b,n$ for the $n$-th prime equal to $b$ modulo $a$ (coprime to $b$)? Bounds on Linnik's constant answer this for the first prime in each ...
8
votes
2answers
391 views

Class number of real maximal subfield of cyclotomic fields

Let $p$ be a prime number and $h_p^+$ the class number of $\mathbb{Q}(\zeta_p + \zeta_p^{-1})$. What is known about the values of $p$ for which $h_p^+ = 1$? Are there infinitely many? Finitely many? ...
8
votes
0answers
365 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, ...
7
votes
1answer
472 views

The Bombieri Vinogradov Theorem restricted to moduli divisible by $k$

The Bombieri-Vinogradov Theorem states that given $A>0$, there exists $B>0$ such that for $Q=\sqrt{x}\left(\log x\right)^{-B},$ we have $$\sum_{q\leq Q}\max_{y\leq x}\max_{\begin{array}{c} ...
2
votes
0answers
119 views

Steps required to recognise a $z$-smooth number

I am currently reading section 5 of Pomerance's paper The Number Field Sieve and I have a few questions about smooth numbers. A number $x\in\mathbb Z_{\ge1}$ is called $z$-smooth if every prime ...
1
vote
0answers
173 views

Proof of the Infinitude of Odd Primitive Pseudoperfect Numbers

I'm interested in the infinitude of odd primitive pseudoperfect numbers. Richard K. Guy's book "Unsolved Problems in Number Theory 3rd edition" says that P. Erdős proved the infinitude of odd ...
1
vote
1answer
84 views

computing $c_5$ in “Primality testing with Gaussian periods”

As far as I know, the April 2011 version of #143 on this page has not been improved upon. On page 10 of that paper, the authors give an algorithm that uses a constant $\:c_{\hspace{.01 in}5}\:$. ...
10
votes
1answer
406 views

Least prime $p$ such that an irreducible polynomial of degree $n$ has no root modulo $p$?

This question is inspired by an old question of Greg Kuperberg, about how small is the first prime $p$ which makes a given monic polynomial $P$ with integral coefficient have a (simple) root modulo ...
2
votes
0answers
143 views

Best known Upper bound on Twin Primes [duplicate]

I know that there is a result from J Wu that the number of twin primes less than a given magnitude $N$ does not exceed $$\frac{2aCN}{\log^2{N}}$$ Where $C=\prod \frac{p(p-2)}{(p-1)^2}$ and $a$ is ...
33
votes
2answers
2k views

A question on maps from $\mathbb{Z}/p\mathbb{Z}$ to itself

Let $p\geq 3$ be a prime number, and let $u:\mathbb{Z}/p\mathbb{Z}\to \mathbb{Z}/p\mathbb{Z}$ be a map such that, for all $l\in \mathbb{Z}/p\mathbb{Z}$,$l\neq 0$, the map $k\mapsto u(k+l)-u(k)$ is a ...
17
votes
1answer
784 views

Infinitely many primes, and Mobius randomness in sparse sets

Problem 1: Find a (not extremely artificial) set A of integers so that for every $n$, $|A\cap [n]| \le n^{0.499}$, ($[n]=\{1,2,...,n\}$,) where you can prove that $A$ contains infinitely many primes. ...
2
votes
1answer
515 views

Prime numbers of a special kind

Is it proved that there is only a finite number of prime numbers of the form $2^{2n} + 2^n + 1$ (where $n \in \mathbb{N}$)?
10
votes
1answer
524 views

A conjecture by Euler about $8n+3$

Euler's conjecture: For any positive integer $n$, $8n+3$ can be represented as a sum $$8n+3=(2k-1)^2+2p,$$ where $k$ is a positive integer, and $p$ is a prime. I want to know whether there has been ...
5
votes
2answers
444 views

General Euclid-Fermat sequences

Question Fermat sequence $$ \forall_{n=1\ 2\ \ldots}\quad F_n:=2^{2^{n-1}}+1$$ has every two different terms relatively prime, and   $1$   does not appear as any of them. Thus it would be ...
12
votes
2answers
730 views

An Euler-proof that cannot be repaired?

Ed Sandifer writes on Euler's wonderful work on prime numbers: The sum of the series of reciprocals of prime numbers $\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+...$ is infinitely large, and is infinitely ...
8
votes
1answer
619 views

Density of prime pairs whose gap is less than the average gap

By the prime number theorem we know that the "average gap" between the first $n$ primes is $\ln p_n$. I would like to know the density of consecutive prime pairs whose gap is less than the average gap ...
1
vote
0answers
130 views

Division between deformed Gausian integers

Let $P_3=${$z\in Z[\sqrt{-3}],|z|^2 \text { is a prime number}, >3$} Let $\alpha_1,...\alpha_n$ be distinct elements in $P_3$, and $l_1,...l_n\in Z^+$. Set ...
2
votes
0answers
321 views

On the primitive prime divisors of $q^n-1$

Let $q=p^\alpha$ be a prime power. We call $r$ a primitive prime divisor of $q^n-1$ where $r\mid (q^n-1)$ but $r\nmid (q^i-1)$ for each $1\leq i\leq n-1$. The set of all primitive prime divisors of ...
1
vote
1answer
92 views

On the set of divisors of $q-1$ and $q'-1$

For a natural number $n$ we denote by $\pi(n)$, for the set of prime divisors of $n$. Let $q=p^\alpha$ and $q'=r^\beta$ where $r$ and $p$ are odd prime numbers and $\alpha,\beta$ are natural numbers. ...
0
votes
1answer
225 views

For any n and some prime p there is an elemnet in Zp* of order n [closed]

How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z_p^*$ contains an element $a$ of order $n$? No ideas at all...
6
votes
0answers
610 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, ...
-1
votes
1answer
339 views

Mersenne Prime Sequences

Hi. Given the following sequence (of Mersenne primes): $ A_{1} = 2 $ $ A_{n} = 2^{A_{n-1}} - 1 $ The first five elements are all prime numbers: $ 2 $ $ 2^{2}-1=3 $ $ 2^{3}-1=7 $ $ 2^{7}-1=127 ...
2
votes
2answers
818 views

Finding primes using Euler's sum of divisors recurrence relation

Euler came up with following recurrence relation for the sum of divisors (refer to http://arxiv.org/abs/math/0411587) $$\sigma(n) = \sigma(n−1) + \sigma(n−2) − \sigma(n−5) − \sigma(n−7) \dots$$ Since ...
3
votes
1answer
336 views

The Nth number with M prime factors

Hi. Suppose we arrange all natural numbers in a matrix P defined as follows: P[I][J] = The Jth number with I prime factors. So P looks something like: 1 2 , 3 , 5 , 7 , 11 , 13 , 17 ...
0
votes
0answers
137 views

Prime-Counting Function

Would a summation of floor[cos^2π(((j-1)!+1)/j)] from j=1 to x be the same as π(x)?
1
vote
1answer
162 views

Behavior of a quantity related to Fermat's 4n + 1 Theorem

One of Fermat's theorems states that if $p = 4n + 1$ for some integer $n$, then $p$ can be expressed uniquely as a sum of two squares, $p = a^2 + b^2$. I am working on a problem and I would like to ...
3
votes
1answer
258 views

Primes in short intervals with a preassigned frobenius

Edited after mistake in the first version. It is known since Selberg that under the Riemann Hypothesis, given an $\epsilon>0$, there is a prime between $x$ and $x+O(x^\epsilon)$ for all $x$ in a ...
8
votes
0answers
698 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, ...
1
vote
1answer
1k views

The Jones-Sato-Wada-Wiens polynomial for prime numbers and differential calculus?

After works of Davis, Matijasevic, Putman and Robinson between 1960 and 1970, we know that every recursively enumerable set of numbers can be represented by a polynomial. In particular, it's the case ...
3
votes
1answer
378 views

Heuristic for Montgomery's conjecture

This is my third question on this site regarding Montgomery's conjecture -- and I apologize if this is too much -- but I am still not understanding well why this conjecture is believed to be true. ...
64
votes
6answers
7k views

Does Zhang's theorem generalize to $3$ or more primes in an interval of fixed length?

Let $p_n$ be the $n$-th prime number, as usual: $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, $p_4 = 7$, etc. For $k=1,2,3,\ldots$, define $$ g_k = \liminf_{n \rightarrow \infty} (p_{n+k} - p_n). $$ Thus the twin ...
17
votes
1answer
4k views

Tightening Zhang's bound [closed]

Inspired by a blogpost by Scott Morrison and ongoing discussion there I decided to create this community wiki to track progress on the original bound of Yitan Zhang. The original bound was ...
4
votes
1answer
593 views

Goldbach's conjecture and Euler's idoneal numbers

Recently, I stumbled upon an interesting statement regarding Quadratic Forms. It is quite well-known and, as I will describe briefly, equivalent to Goldbach's conjecture. Let $p,q$ be odd primes and ...
20
votes
4answers
4k views

How does Yitang Zhang use Cauchy's inequality and Theorem 2 to obtain the error term coming from the $S_2$ sum

I have been reading Yitang Zhang's paper now for one and a half weeks and also volunteered to give a popular talk on the paper next week at Stockholm University. Today I found a detail in the proof ...
12
votes
2answers
1k views

Distinctive property of the primes 17 and 19?

Consider the question whether it is true that a prime number $p$ divides $1^1+2^2+3^3+....+(p-1)^{p-1}$ if and only if $p \in \{17,19\}$. For the obvious heuristic reasons, for large $n$ one would ...