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 Riemann's explicit formula imply invariance of the prime gaps distribution under a Fourier-like transform?

Loosely speaking, Riemann's explicit formula states that there exists a Fourier-type duality between the primes and the non trivial zeroes of the Riemann zeta function. Does this mean that the ...
5
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1answer
415 views

Has this strengthening of the PNT already been conjectured?

Suppose $f:\mathbb{Z}_{\geq 0}\to\mathbb{Z}_{\geq 0}$ is an arithmetic function that grows slower than the identity map. Has it already been conjectured that, under this general hypotheses, ...
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0answers
73 views

Where to include contact details in math paper? [on hold]

I recently submitted a paper to a math journal on a prime number patter using latex formatting, but they sent an email back saying that the contact details for the corresponding author should be in ...
26
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1answer
2k 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 ...
2
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1answer
386 views

On quantities with no very small odd prime factors; a response to Wlodzimierz Holsztynski

In response to a comment posted under Powers of $2$ and the products of initial odd primes , I shall raise some questions about quantities near $O_n= P_{n+1}/2$, the product of the first $n$ odd ...
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3answers
209 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 ...
9
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1answer
353 views

Roots of unity near 1 in Z / p Z

Let $r \ge 3$ be a fixed integer. I'm interested in primes p such that no integer in the interval $(-\sqrt{p}, \sqrt{p})$, except $1$ (and $-1$ if $r$ is even), is an r-th root of unity modulo p. The ...
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2answers
404 views

Yitang Zhang's paper [closed]

I just want make thing clear for myself. Others may have asked before in different ways. Does Yitang Zhang's paper prove that for any given length gap $g_n > N$ there is a prime $p_n$ for which ...
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77 views

The sum of the reciprocals of the odd primes taken two at a time [closed]

From Steven R Finch, Mathematical Constants, page 95: The sum of the squared reciprocals of primes is 0.4522474200… Clearly, the sum of the reciprocals of the odd primes taken two at a time -- ...
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0answers
137 views

OEIS A182514 conjectures? [closed]

I have shown in sequence A182514 "Primes prime(n) such that (prime(n+1)/prime(n))^n > n." at https://oeis.org/A182514 that 1693182318746371 is a(6), by using maximal prime gaps. Now, I am wondering if ...
2
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1answer
240 views

Looking for paper: Weil's original 1952 “Sur les formules explicites de la théorie des nombres premiers”

I am looking for a source (preferably online) for Weil's original 1952 paper on the explicit formula. I am aware of an english translation available here, but would like to have access to the original ...
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1answer
136 views

Congruences among primes modulo which a given polynomial has roots

Suppose $f(x)\in\mathbf Z[x]$ is nonconstant. I would like to know if either of the following statements is true. If $a$ and $b$ are coprime integers (probably with some additional restriction), ...
6
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0answers
143 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 ...
3
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0answers
135 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 + ...
5
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0answers
208 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 ...
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1answer
148 views

First Parameterized Subset of Primes that was Related to a Mathematical Result

To my knowledge, Fermat primes, i.e. primes of the form $2^{2^n}+1$ were the first to play a role in a mathematical result, namely in the characterization of constructible regular n-gons. Gauss ...
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1answer
426 views

What keeps asymptotic Goldbach's conjecture out of reach of current technology?

Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...
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244 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? ...
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0answers
110 views

Lehmer's Totient Problem

Recently while researching on the famous Lehmer's Totient Problem I have found that only counterexamples can arise from the following conditions- Let $n$ be a odd positive Carmichael Number and let ...
3
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1answer
104 views

Decidability of prime gap sequences

Is the following problem undecidable? Given a sequence of $n$ gaps $d_1,d_2,...,d_n$, does there exist a sequence of $n+1$ primes $p_1,p_2,...,p_{n+1}$ such that $p_{i+1} - p_i = d_i$ ? If not, is ...
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0answers
143 views

Average order and upper bound of $r_{0}(n)$

Assume Goldbach's conjecture. Then for every integer $n>1$ there exists a non-negative integer $r$ such that $n-r$ and $n+r$ are both primes. For a given $n>1$, the smallest such $r$ will be ...
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4answers
3k 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 $d$ ...
6
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1answer
199 views

Question about a certain class of primes

I've come across a set of primes in a problem I'm working on, and I'm wondering if there's more information available about them. I'm guessing not much, particularly since the question of infinitude ...
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0answers
174 views

An inequality about Goldbach conjecture

Let $N$ a large natural number, let $\forall n\leq N,\, R_{2}\left(n\right)=\underset{p_{1}+p_{2}=n}{\sum}\log\left(p_{1}\right)\log\left(p_{2}\right)$ and let $S\left(\alpha\right)=\underset{p\leq ...
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2answers
218 views

Infinite play with tape, or covering the integers with prime arithmetic progressions

It is possible that a more technical version of this question has been asked and answered in the literature. If so, then a reference is much appreciated. I will phrase it in terms of colored tapes ...
2
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1answer
70 views

Principally split primes with factors in arbitrarily small angular sectors

I wonder if the following is known: let $n$ be a (square-free) positive integer. Is there ever/always a sequence of prime numbers $p$ that can be written in the form $$p = x^2 + ny^2,$$ where $x, y$ ...
3
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0answers
127 views

An estimate for dividing n^2 by each of the primes up to and including n, and then summing the results [closed]

I know that the asymptotic for the sum of all the primes up to n is $n^2/2\log n$. But I'm trying to find the formula (an estimate) for when $n^2$ is divided by each of the primes up to $n$, in turn ...
3
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1answer
341 views

When does Merten's product theorem accurately estimate the number of coprimes in an interval?

Assume an arbitrary $x$ and let $z$ be smaller than $y$, where $y$ is the length of the interval $[x,x+y]$. What I would like to know is: Let $W(z)=\prod_{p\leq z}\left(1-\frac{1}{p}\right)$. For ...
5
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2answers
219 views

Sum of digits of repeating end of reciprocal of prime over period is $\frac{9}{2}$

Take a prime other than 2,3 or 5 and look at the part of it that repeats in base 10. Is it true that the sum of the digits in the end divided by the period(number of repeated digits id always ...
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3answers
290 views

Does this 'alternating' Euler product converge for all $\Re(s) > 0$?

Does the following 'alternating' Euler product, with $p_n$ the $n$-th prime number, converge for $\Re(s)>0$ ? $$\displaystyle \prod_{n=1}^\infty \left( \dfrac{1}{1-\frac{1}{p_{n}^{s}}} ...
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3answers
525 views

Definition of Prime Numbers [duplicate]

The first time I heard of prime numbers, they were defined as natural numbers $n$ that can only be divided by 1 and themselves without remainder; later, when prime factorization was introduced, I ...
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2answers
207 views

What is the minimal range $[f(n),g(n)]$ that contains a prime number for every integer $n>0$?

I know the following: Proven: There is a prime number between $n$ and $2n$ for every integer $n>0$. Conjectured: There is a prime number between $n^2$ and $(n+1)^2$ for every integer $n>0$. ...
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2answers
215 views

prime zeta function when $0<s<1$

I will not be surprised if this question seems trivial in MO but i asked it first in MathSE and i did not get an answer. So, here it is: I would like to know if there is a good estimate for the sum ...
4
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1answer
400 views

Lines in image; are they significant to prime numbers if so how?

Amateur math question. I was playing around generating some 2D images, and wondered what it would look like if placed $P_{i}$ dots on a circle with diameter of $i$ for increasing values of $i$, where ...
3
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1answer
303 views

Giuga's Conjecture: Central or Peripheral?

An earlier MO question highlighted Giuga's Conjecture: A positive integer $n>1$ is prime if and only if $$\sum_{k=1}^{n-1} k^{n-1} \equiv -1 \pmod{n}$$ For example, for the prime $n=5$, ...
3
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0answers
173 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 ...
10
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1answer
389 views

Chebotarev density theorem for $k$-almost primes

Consider a finite Galois extension $L$ of $\mathbb Q$, of Galois group $G$. Let $k \geq 1$ be a fixed integer. Let $D$ be a subset of $G^k$ invariant by conjugation and by the natural action of the ...
4
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1answer
309 views

Does this prime-gaps pattern occur infinitely often?

Let $p_n$ be the $n$-th prime. For each integer $k \ge 0$, do there exist an infinite number of $k+3$ consecutive primes $(p_n, p_{n+1}, \ldots, p_{n+2+k})$ so that (1) The gap between the 1st and ...
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1answer
179 views

Giuga and Carmichael numbers

If $p$ is both Giuga and Carmichael number then its known that $1^{p-1}+2^{p-1}+3^{p-1}+\cdots+(p-1)^{p-1} \equiv -1\pmod{p}$ is it true that if $p$ is both Giuga and Carmichael number then ...
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0answers
141 views

Interval containing prime numbers

Let $\varepsilon$ be an arbitrary small positive number. Can we prove that there exist an $n\in Z$ such that the interval $[2^n,(1+\varepsilon)2^n]$ contain a prime number?
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226 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 ...
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1answer
97 views

Fermat pseudo prime base-3 [closed]

Good morning! I have checked the following statement by random numbers of my choice. I am seriously looking for proof of the statement. Statement: $m$ is said to be Fermat pseudo prime in base-3, ...
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0answers
116 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 ...
5
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3answers
301 views

Weak versions of Bertrand's postulate

We are interested in the following statement: For each $n>1$ and $x>2$ there is at least one prime $p$ satisfying $x<p<n x$. For $n=2$ we get precisely the Bertrand's postulate which is ...
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0answers
41 views

existence of elements with specific norms in pure cubic fields

Is there any specific way to find an element with a given norm in pure cubic field? say for an example an element of norm 5 in pure (monogenic) cubic field of 11. it is easy to check that 5 as an ...
4
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1answer
143 views

Consecutive Primes mod 3

Is anything known asymptotically about the binary "primes mod 3" sequence besides Dirichlet's result that 1 and 2 occur half of the time? For example, can you prove that it does not eventually cycle ...
4
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2answers
281 views

Can a polynomial be almost always divisible by a member of a finite set of primes?

Special case of Bunyakovsky conjecture Let $f(x)$ be non-constant irreducible polynomial with integer coefficients, no fixed prime factor and positive leading coefficient. Let $S$ be a finite set of ...
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1answer
80 views

Joint Modular Distribution of Primes

Dirichlet's theorem shows that, for any fixed prime integer a, "big prime numbers mod a" are uniformly distributed between 1 and a-1. If we similarly pick different prime integers b,c,..., are these ...
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115 views

Updated tables of maximal prime gaps? [closed]

The website http://www.trnicely.net/gaps/gaplist.html contains a long list of maximal and nonmaximal prime gaps. In this list, the largest maximal prime gap is one of length 1476: Size: 1476 Gap ...
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1answer
290 views

Are primes of density 0 in $a\cdot b^n+c$?

Hooley proves in Applications of Sieves to the Theory of Numbers that there are only $o(x)$ numbers $n\le x$ such that $n\cdot2^n+1$ is a (Cullen) prime. The proof generalizes to forms ...