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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3
votes
0answers
314 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 ...
9
votes
1answer
258 views

Repetend digit graphs for $1/n$ in base $b$

Here is a decimal expansion of $\frac{1}{34}$: $$(1/34)_{10}=0.02941176470588235\overline{2941176470588235}\ldots$$ And here is a graphical representation of the 16-digit "repetend," as a directed ...
3
votes
1answer
348 views

Lower bounds on the error term of the prime number theorem

Are there any lower bounds on the error term for the prime number theorem, or in other words, is there a nontrivial $f$ s.t. $$f(x)\ll |\psi(x) - x|$$ where $\psi$ is the Chebyshev function.
3
votes
1answer
386 views

Smallest prime in an arithmetic progression

Let $\{a_n\}_{n\in\mathbb{N}}$ be defined as $a_n = a + bn$ for some $a, b >0,(a, b) = 1$. Are there good bounds on the minimal $k$ s.t. $a_k$ is prime. It is well known that there are infinitely ...
8
votes
1answer
230 views

Integers with a large prime divisor in short intervals

For an integer $n$, denote by $P^+(n)$ the largest prime divisor of $n$. Then we have the following: There exists some $c>0$, such that for all $x$ sufficiently large the number of integers ...
7
votes
1answer
242 views

lower and upper bound for $\sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k$?

Are there known any lower and upper bounds for $$ \sum_{k=1}^n \frac{(-1)^{\Omega(k)}}k, $$ where $\Omega(n)$ is the number of prime factors counting multiplicities of $n$? Or at least is it known ...
6
votes
2answers
248 views

Divisor sums over values of binary forms of primes

Let $\tau$ be the divisor function, that is $$ \tau(n)=\sharp\{d \in \mathbb{N}, d|n\}. $$ I was wondering if anyone has ever proved an asymptotic estimate for the sum $$S(x):=\sum_{p,q\leq ...
0
votes
1answer
175 views

Conjectured Primality Test for Numbers of the Form $k \cdot 2^n+1$ with $n>2$ [closed]

Definition : Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^m+\left(x+\sqrt{x^2-4}\right)^m\right) $ where $m$ and $x$ are positive integers . Conjecture : Let $N=k\cdot 2^n+1$ with ...
10
votes
1answer
321 views

Squarefree numbers $n$ such that $432n+1$ is also squarefree

This is a second attempt (see Primes $p$ such that $432 p +1$ is prime) Is the set of squarefree numbers $n$ such that $n(432 n+1)$ is also squarefree known to be infinite? Fact: the number of such ...
-1
votes
1answer
310 views

Primes $p$ such that $432 p +1$ is prime [closed]

Is the set of prime numbers $p$ such that $432 p + 1$ is also prime infinite? It doesn't follow from Dirichlet's theorem as far as I can tell.
5
votes
2answers
654 views

The shortest interval for which the prime number theorem holds [closed]

It is well known that the prime number theorem on the form \begin{align*} \pi(x+y) - \pi(x) \sim \frac{y}{\log (x+y)} \end{align*} breaks down for short enough intervals, e.g. taking $y=(\log ...
0
votes
0answers
123 views

Does $\pi(n+r)+\pi(n-r)$ decrease as $r$ increases?

Assume Goldbach's conjecture. Then for every large enough positive integer $n$ there exists a non negative integer $r$ such that both $n+r$ and $n-r$ are primes. Such an integer $r$ will be called a ...
6
votes
1answer
315 views

is there any heuristics suggesting that the number of Fibonacci primes below $x$ is equivalent to $\log_{\phi}\log_{\phi}x$?

The question of knowing whether there are infinitely many Fibonacci primes is an open question. As $F_p$ is prime only if $p$ is prime, one has $\pi_{FP}(x)\le \pi(\log_{\phi} x+0.5\log 5)$, but ...
14
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 ...
3
votes
1answer
307 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 ...
13
votes
1answer
690 views

Tight prime bounds

This is a cross-post of this question on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the ...
0
votes
0answers
115 views

When is the earliest large prime gap also the latest large prime gap?

Suppose one finds the earliest prime gap of at least a certain size $g$, so that $p_{n+1}-p_n=g$ and $n$ is the smallest index for which the gap is as big as $g$. Now consider the relative size of ...
1
vote
2answers
229 views

overlap quadratic residues

Let $p$ be a prime number of form $4k+1$ and $M$ is its quadratic residue set. Let $M_i=\{i+x|\forall x\in M\}$ $\forall 0<i<p$. Does there exist a positive constant $\varepsilon$ such that ...
4
votes
1answer
211 views

How frequently is 3 a cubic residue mod primes in an arithmetic progression?

Suppose $(a,3q)=1$ and $a\equiv 1\pmod 3$. Are there infinitely many primes $p\equiv a\pmod {3q}$ such that $3$ is a cubic nonresidue modulo $p$? Or, an equivalent formulation using quadratic forms: ...
6
votes
1answer
510 views

Is this weak asymptotic Goldbach's conjecture open?

Let $\tau(x)$ be the number of even numbers $2<2n<x$ which can't be written as a sum of two primes. Goldbach's conjecture: $\tau(x) = 0$ Asymptotic Goldbach's conjecture: $\tau(x) = O(1) $ ...
0
votes
2answers
238 views

Goldbach-type problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number. It follows that $\phi(\prod_i p_i^{n_i}) = \sum_i n_i p_i$, with $p_i$ a prime ...
2
votes
0answers
171 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 $$ ...
17
votes
1answer
706 views

The conjecture of Montgomery and Soundararajan on primes in short intervals: Empirical inconsistencies?

Assume that $y/ \log x \rightarrow \infty$ and that $y/x \rightarrow 0$. Then, from a conjecture by Montgomery and Soundararajan, we expect the number of primes in the interval $[x,x+y]$ to be ...
0
votes
0answers
284 views

Arithmetic progression and average of two prime numbers

Let $A=(a_n : n \in \mathbb{N})$ be the sequence given by: $$ \ a_n = a_1 + (n - 1)d,\quad a_1,\ d,\ n \in \mathbb N,\quad d\gt a_1,\quad \gcd(a_1,\ d)=1. $$ For all terms of $A$ greater than $\ ...
2
votes
0answers
105 views

Primality Criterion for Specific Class of Numbers of the Form $k\cdot b^n-1$

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
95 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 ...
3
votes
2answers
415 views

Primes from a Dirichlet sequence and an irrational number

From Dirichlet's theorem on arithmetic progressions, if $\text{gcd}(a,b)=1$ we know $\{ak+b\}_{k\ge 0}$ contains infinitely many primes. Let those primes be $p_1,p_2,\cdots$. Then the real ...
0
votes
0answers
59 views

Prime Hadamard Matrices

Assume that $n$ is a sufficiently large number. Is there a Hadamaed matrix $H_{4n \times 4n}=(h_{ij})$ with the last row and the last cloumn $J$ (thet is, for every $k$, $h_{k,4n}=1$ and $h_{4n, ...
10
votes
1answer
370 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
4answers
1k views

Arbitrarily long arithmetic progressions

Are there arbitrarily long arithmetic progressions in which all the prime factors of all the terms are at most $N$, for some $N$? Assume all the terms are positive and the sequence of terms is ...
5
votes
2answers
196 views

Relationship of Euler product to coprimality densities for arbitrary sets of primes

Continuing the curiosity of my last couple questions: Is it the case that for every set of primes $F$, the asymptotic density of the integers coprime to all of $F$ is $\displaystyle \prod_{p \in F} (1 ...
2
votes
1answer
312 views

Finding a suitable number

Let $n,m$ be two positive integers. By $r_n$ we denote the largest prime not exceeding $n$. If $r_n\leq m\leq n$ and $q$ is the largest prime factor of $n!/m!$ such that $q\geq 17$ and $q\geq n-m+3$, ...
1
vote
2answers
233 views

Consecutive primes versus prime twins

First a warm-up. Let $\ V\ $ be an arbitrary set of odd natural numbers. Let $\ S(V)\ $ be the generated multiplicative semi-group. What are the necessary and/or sufficient conditions on $\ V\ $ for ...
6
votes
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
157 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
170 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
186 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 ...
5
votes
0answers
238 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
454 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
379 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
203 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
209 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
364 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
226 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
318 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
125 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
116 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
365 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 ...