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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7
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3answers
718 views

Bound the error in estimating a relative totient function

Let $n=p_1^{e_1}\cdots p_k^{e_k}$ be an integer with $k$ prime factors. We know that the number of integers less than $n$ and coprime to it is $$\Phi(n)=n-\sum_i\frac n{p_i}+\sum_{i \lt j}\frac ...
9
votes
0answers
776 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, ...
19
votes
7answers
3k views

Asymptotic density of k-almost primes

Let $\pi_k(x)=|\{n\le x:n=p_1p_2\cdots p_k\}|$ be the counting function for the k-almost primes, generalizing $\pi(x)=\pi_1(x)$. A result of Landau is $$\pi_k(x)\sim\frac{x(\log\log ...
13
votes
2answers
635 views

Odd-bit primes ratio

Say that a number is an odd-bit number if the count of 1-bits in its binary representation is odd. Define an even-bit number analogously. Thus $541 = 1000011101_2$ is an odd-bit number, and $523 = ...
0
votes
1answer
483 views

A possible consequence of Dirichlet's theorem about primes in arithmetic progression

EDIT : I copy-paste the beginning of a previous question since Gerry Myerson suggested this question should be self-contained. "let's consider a composite natural number $n$ greater or equal to $4$. ...
4
votes
1answer
509 views

Lower bound for a prime gap occurring infinitely often

In his striking paper of may 2013, Zhang showed the existence of an even integer $g\lt 70,000,000$ such that $g$ is a prime gap occurring infinitely often. What is the best unconditional lower bound ...
33
votes
4answers
2k views

Are There Primes of Every Hamming Weight?

That is, for every integer $n \in \mathbb{Z}_{>0}$ does there exist a prime which is the sum of $n$ distinct powers of $2$? In this case, the Hamming weight of a number is the number of $1$s in ...
34
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 ...
7
votes
2answers
1k views

least prime in a arithmetic progression

Hello Here I want to consider the simplest arithmetic progression $n\equiv 1\pmod{q}$ where $q$ is a prime. Is it true that we can find a prime $p\leq q^2$ in this arithmetic progression? This ...
1
vote
3answers
257 views

Powers of $2$ and the products of initial odd primes

NOTATION: $O_x$ -- the product of all odd primes $\le x$. E.g. $O_7=3\cdot 5\cdot 7 = 105$. QUESTION: Are the three ordered pairs $\ (d\ p)=(1\ 3)\ \ (2\ 3)\ \ (4\ 5)\ $ the only solutions of the ...
42
votes
1answer
9k views

Is the Green-Tao theorem true for primes within a given arithmetic progression?

Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes. Now, consider an arithmetic progression with starting term $a$ and common difference $d$. ...
65
votes
6answers
8k 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 ...
21
votes
4answers
2k views

Why so difficult to prove infinitely many restricted primes?

I wondered whether there were an infinite number of palindromic primes written in binary (11, 101, 111, 10001, 11111, 1001001, 1101011, ...) and quickly discovered that it is unknown (OEIS A117697). ...
15
votes
4answers
1k views

Arithmetic progressions without small primes

The following question came up in the discussion at How small can a group with an n-dimensional irreducible complex representation be? : Is it known that there are infinitely many primes p for which ...
22
votes
4answers
1k views

Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...
6
votes
2answers
658 views

At what point would an elementary generalization of Bertrand's Postulate be interesting?

I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$. At what point would an improvement on Nagura's result be interesting? ...
7
votes
2answers
973 views

What is the lower bound for highly composite numbers?

if $x=d(n)$ is the number of divisors of $n$, what is the tightest lower-bound for $n$ only given $x$? http://en.wikipedia.org/wiki/Highly_composite_number
10
votes
1answer
429 views

Primes dividing $2^a+2^b-1$

From Fermat's little theorem we know that every odd prime $p$ divides $2^a-1$ with $a=p-1$. Is it possible to prove that there are infinitely many primes not dividing $2^a+2^b-1$? (With ...
7
votes
1answer
1k views

What would be the consequences of $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}$?

The question is in the title: what would be the number theoretic consequences if we managed to establish the conjectured asymptotic equality $\displaystyle{\lim\inf_{n\to\infty}p_{n+k}-p_{n}\sim k\log ...
6
votes
3answers
308 views

Quadratic residues and nonresidues of arbitrary patterns

Let $p_1, p_2, \dotsc, p_n$ be distinct primes, and let $\epsilon_1, \epsilon_2, \dotsc, \epsilon_n$ be an arbitrary sequence of $1$ and $-1$. There is an integer $a$ such that $\left( \frac{a}{p_1} ...
6
votes
1answer
994 views

Number field analogue of the Goldbach Conjecture

Is there a generalization of Goldbachs conjecture for prime ideals in number fields?
3
votes
3answers
3k views

Algorithm for detecting prime powers

While reading Peter Shor's paper Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer, I came across the following quote: "This scheme will thus work as ...
11
votes
1answer
954 views

Order of magnitude of $\sum \frac{1}{\log{p}}$

Question: What is the order of magnitude of the following sum? $$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$ Additional information: Since $$ ...
10
votes
2answers
891 views

Update for 2015: least prime of form nq+1, with q prime?

I have received a complaint about my 2011 answer least prime in a arithmetic progression which, indeed, gives conflicting reports about this: given a prime $q,$ what can we say about an upper ...
10
votes
2answers
1k views

Natural numbers that cannot be expressed as a difference between a square and a prime?

We wish to find the set of natural numbers that cannot be expressed as a difference between a square and a prime. e.g. $1 = 2^2 - 3$ $2 = 3^2 - 7$ $3 = 4^2 - 13$ and so on. The smallest such ...
5
votes
1answer
414 views

Unexpectedly prime rich cubic polynomial

We got a cubic polynomial which is unexpectedly prime rich. Let $f(x)=29160 x^3 + 30132 x^2 + 8046 x + 643$ and $\pi_f(n)$ the number of primes values of $f(x)$ for $x \in [1,n]$. Let ...
7
votes
2answers
443 views

What is wrong with this deterministic algorithm efficiently generating large primes?

According to PolyMath (Strong) conjecture. There exists deterministic algorithm which, when given an integer k, is guaranteed to find a prime of at least k digits in length of time polynomial in ...
6
votes
0answers
643 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, ...
3
votes
0answers
329 views

Metric on the set of subsets of the rational primes

Note: this is a revision of an earlier post. It was kindly pointed out that my initial proposed metric was in fact not a metric, so this is a revised version. I was thinking how to say that two sets ...
3
votes
2answers
386 views

Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?

OEIS A226181: 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ... Primes $p$ ...
3
votes
3answers
561 views

Estimate for products of integers that are relatively prime with $N$

Let $N$ be a positive integer. Are there known estimates for the product of all numbers that are smaller than $N$ and relatively prime with $N$? One can assume that $N$ is free of squares, if this ...
-3
votes
2answers
302 views

The number of totatives to the nth primorial, in an interval shorter than the nth primorial

(The notation of this question will be improved over the next few days, sorry for the lack of clarity at the moment.) Can, and if so when can, we determine the amount of natural numbers which are ...
81
votes
4answers
30k views

Philosophy behind Yitang Zhang's work on the Twin Primes Conjecture

Yitang Zhang recently published a new attack on the Twin Primes Conjecture. Quoting Andre Granville : “The big experts in the field had already tried to make this approach work,” Granville ...
95
votes
5answers
6k views

Gaussian prime spirals

Imagine a particle in the complex plane, starting at $c_0$, a Gaussian integer, moving initially $\pm$ in the horizontal or vertical directions. When it hits a Gaussian prime, it turns left ...
34
votes
4answers
4k views

Why could Mertens not prove the prime number theorem?

We know that $$ \sum_{n \le x}\frac{1}{n\ln n} = \ln\ln x + c_1 + O(1/x) $$ where $c_1$ is a constant. Again Mertens' theorem says that the primes $p$ satisfy $$ \sum_{p \le x}\frac{1}{p} = \ln\ln ...
41
votes
4answers
10k views

How hard is it to compute the number of prime factors of a given integer?

I asked a related question on this mathoverflow thread. That question was promptly answered. This is a natural followup question to that one, which I decided to repost since that question is answered. ...
48
votes
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 ...
27
votes
4answers
4k views

What is exceptional about the prime numbers 2 and 3?

Admittedly this question is vague. But I hope to convey my point. Feel free to downvote this. Permit me to define prime number the following way: A number $n>1$ is a prime if all integers $d$ ...
40
votes
1answer
3k views

Are the primes normally distributed? Or is this the Riemann hypothesis?

Forgive my very naive question. I know next to nothing about number theory, but I'm curious about the state of the art on the distribution of primes. Let $\mathrm{Li}(x)$ be the offset logarithmic ...
40
votes
1answer
2k views

Exploding primes

Suppose every prime $n$ could "explode" once. An explosion results in $\lfloor \alpha \ln n \rfloor$ particles being uniformly distributed over the integers in a range $n \pm \lfloor \beta \ln n ...
31
votes
1answer
3k views

Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $

EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems ...
17
votes
3answers
1k views

Prime factorization “demoted” leads to function whose fixed points are primes?

Let $n$ be a natural number whose prime factorization is $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \; .$$ Define a function $g(n)$ as follows $$g(n)=\sum_{i=1}^{k}p_i {\alpha_i} \,$$ i.e., exponentiation is ...
23
votes
2answers
2k views

What is the crucial difference the Maynard/Tao approach and Goldston-Pintz-Yildirim that extends to prime k-tuples with $k>2$

Suppose $m$ is a positive integer. A quantity of interest is $$ H_m = \liminf_{n\to\infty} \left(p_{n+m} - p_n \right) $$ The twin prime conjecture, is, of course $H_1 = 2$, the the prime k-tuples ...
17
votes
6answers
2k views

Question on consecutive integers with similar prime factorizations

Suppose that $n=\prod_{i=1}^{k} p_i^{e_i}$ and $m=\prod_{i=1}^{l} q_i^{f_i}$ are prime factorizations of two positive integers $n$ and $m$, with the primes permuted so that $e_1 \le e_2 \cdots \le ...
11
votes
7answers
2k views

Are there any interesting or lesser known proofs related to Bertrand's Postulate

There are 3 standard proofs of Bertrand's Postulate: (1) Chebyshev's original proof (2) Ramanujan's simplification of Chebyshev's proof (3) Erdos's proof I recently learned about the very ...
16
votes
6answers
10k views

Examples of prime numbers in nature [closed]

Finding primes in signals is seen as a sign of some kind of intelligence - see e.g. the role of primes in the search for extraterrestrial life (see e.g. here). This is because there are relatively few ...
15
votes
1answer
961 views

Sums of primes that are themselves prime

I'm not a math expert so this may be a trivial question; if $p_i$ is the $i$-th prime, let: $$S(n) = \sum_{i=1}^n p_i$$ be the sum of the first $n$ primes and $$P(n) = | \{1 \leq i \leq n \mid ...
24
votes
3answers
2k views

Composite pairs of the form n!-1 and n!+1

It's well known that the numbers of the form $n!\pm1$ are not always prime. Indeed, Wilson's Theorem guarantees that $(p-2)!-1$ and $(p-1)!+1$ are composite for every prime number $p > 5$. Is ...
15
votes
1answer
594 views

A Collatz-like function that bifurcates on primes

This is likely piling one mystery on another, but ... I was exploring a function $f(n): \mathbb{N} \mapsto \mathbb{N}$ defined as follows: $$ f(n) = \begin{cases} n^2 & \text{if} \;n \;\text{is ...
11
votes
1answer
2k views

The Green-Tao theorem and positive binary quadratic forms

Some time ago I asked a question on consecutive numbers represented integrally by an integral positive binary quadratic form. It has occurred to me that, instead, the Green-Tao theorem may include a ...