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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6
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5answers
2k views

Optical methods for number theory?

I found a paper: 'A New Method of Finding the Distribution of Prime Number', saying We stack discs and annuluses with certain rules then turn on the light to illuminate. The projection of ...
4
votes
1answer
165 views

Log weight removal in general (weaker) prime number theorem

Let $a_n$ be a sequence of non-negative numbers. Assume that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} a_p\log p}{X}\leq 1.$$ Can we prove that $$\limsup _{X\to \infty}\frac{\sum_{p\leq X} ...
4
votes
1answer
172 views

Prime residua races and two views on primes

Let $\ a>1\ \ r\ \ k\ $ be arbitrary natural numbers such that $\ a\ r\ $ are relatively prime. The natural conjecture below, is it known?, is probably true in full generality: Q1. There exists a ...
7
votes
0answers
201 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 ...
6
votes
0answers
264 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 ...
3
votes
3answers
486 views

Conjecture about a sequence of natural numbers, such that, $\forall n : A_n<P_n<A_{n+1}$

Conjecture - no natural number $k$ exists such that: $P$ is the sequence of all primes starting from the $k$th prime $A$ is a sequence of natural numbers such that: $\forall n : ...
18
votes
2answers
1k views

The prime numbers modulo $k$, are not periodic

Consider the sequence of prime numbers: $2,3,5,7, \cdots$. Now reduce this sequence modulo $k$ for some integer $k > 2$. Show the resulting sequence is not periodic. : EDIT: As noted in the ...
2
votes
0answers
430 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 ...
0
votes
0answers
206 views

Relationship between this conjecture and Lehmer's Theorem?

Let A be: n such that $\ \frac{n-1}{ord_n 2}=2^x\ $ and $n$ with the conditions of the conjecture in OEIS A226014,$\ n \in \mathbb{Z^+} ,\ x \in \mathbb{Z}_{\geq 0},\ $then $n$ is prime ...
5
votes
1answer
229 views

Uniformity of the distribution of the prime numbers on the prime residue classes (mod $m$)

Given positive integers $m$, $r$ and $n$, let $\pi(m,r,n)$ denote the number of prime numbers $p \leq n$ in the residue class $r$ (mod $m$). Further let $1 = r_1 < r_2 < \dots < ...
3
votes
2answers
372 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 $p$ ...
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 ...
2
votes
0answers
247 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 ...
0
votes
1answer
322 views

Is a certain sumset derived from primes of a certain form the set of all naturals?

OEIS A167055 Numbers n such that $12n + 5$ is prime. $0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS $A167055$. I conjecture that the set of the sum of every two items of this ...
2
votes
0answers
136 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 ...
1
vote
0answers
145 views

Prime counting function with a form of finite product using perron's formula

There's a form of complex integral what Riemann obtained to finding $\pi (x)$, $$ \pi^{*}(x)=\int_{L}\frac{\log \zeta (s)}{s}x^{s}ds, (1)$$ we already know that it lead us to the Prime Number ...
2
votes
1answer
390 views

Number of primes with $-1\pmod 6$ vs. Number of primes with $+1\pmod 6$

Have not been able to get an answer to this on http://math.stackexchange.com, so trying here too... Given the following two sets: $P^-(n) = \{p \leq n : p \equiv -1\pmod 6\}$ $P^+(n) = \{p \leq n ...
0
votes
0answers
114 views

Can the approach followed in this article be used to improve the upper bounds for $H_{k},k>1$?

In http://arxiv.org/pdf/1405.0682.pdf, the author gives a conditional proof of the twin prime conjecture under both a generalized version of the Elliott-Halberstam conjecture and a hypothesis on the ...
3
votes
0answers
112 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 ...
7
votes
1answer
437 views

Major arcs in the proof that every odd number is the sum of at most 5 primes

In his proof that all odd numbers greater than 1 are the sum of at most 5 primes, Terence Tao uses one large major arc around 0 rather than small ones around the rationals, which I am more accustomed ...
1
vote
0answers
151 views

Identity on sum over reciprocals of prime products?

The following identity seems to follow from a simple analysis of the sieve of Eratosthenes and inclusion-exclusion, where $p_i, p_j, p_k, \ldots$ denote primes and $N$ is an integer $\geq 2$: ...
4
votes
2answers
212 views

Orders of the conjugates of an algebraic prime number in its residue field

Of interest to me is the following question (it would be nice to find out what is known in its direction): Given a Galois number field $K/\mathbb{Q}$ and a completely and principally split prime ...
1
vote
0answers
74 views

Error analysis needed for more refined estimates (than Salat-Zanam) of the sum of prime powers

A recent question on math overflow on sums of the primes squared was answered/put on hold by pointing to an old paper by T. Salát and S. Znám, On the sums of the prime powers. Salat and Znam's (SZ) ...
1
vote
1answer
182 views

A square-squareroot integer race sequence involving primes

I wonder what is the expected behavior of this process? Let $f^2_{\mathrm{next}}(n) =$ the next prime after $n^2$. $g_{\mathrm{sqrt}}(n) = \lfloor \sqrt{n} \rfloor$. Now iterate as ...
3
votes
1answer
184 views

Wiener-Ikehara tauberian theorem and order of pole at s=1

In the introduction to Akshay Venkatesh's thesis "Limiting Forms of the Trace Formula" we have the following statement : "For, in summing over primes, the limit ...
2
votes
1answer
184 views

About a Variant of Ulam Spiral

I asked this on Math.SE but got no answer: Here I read about a variant on the Ulam spiral: [A] structure may be seen when composite numbers are also included in the Ulam spiral. [...] Using ...
2
votes
1answer
220 views

Number theoretic functions that have an irregular behaviour at primes

Usually, number theoretic functions have "trivial" (or at least easily defined) values for primes. In this thread, I am rather asking for functions which are only defined on primes (well, this ...
8
votes
1answer
381 views

Prime races à la Mertens

I have just read the nice survey by Granville and Martin about prime races. I wonder what happens if one changes the rules for the prime races as follows. Fix $q$ a modulus (an integer $>1$). For ...
2
votes
1answer
313 views

Is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture?

The question is in the title: is Hardy-Littlewood k-tuple conjecture known to imply Goldbach's conjecture? I tried to give a heuristics in Upper bound for $r_{0}(n)$ through probabilities that seems ...
5
votes
1answer
501 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, ...
39
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 ...
2
votes
1answer
445 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 ...
1
vote
3answers
234 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 ...
11
votes
1answer
418 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 ...
0
votes
2answers
790 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 ...
2
votes
1answer
354 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 ...
3
votes
1answer
174 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), ...
10
votes
1answer
376 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 ...
4
votes
0answers
170 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
votes
0answers
252 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 ...
1
vote
1answer
162 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 ...
7
votes
1answer
625 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 ...
6
votes
0answers
319 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? ...
3
votes
1answer
151 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 ...
27
votes
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
votes
1answer
218 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 ...
1
vote
0answers
216 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 ...
1
vote
2answers
250 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
votes
1answer
85 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
votes
0answers
129 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 ...