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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218 views

Is this a proof of the Hardy-Littlewood inequality? [closed]

V.V. Miasoyedov posted a paper to the arXiv claiming a proof of the Hardy-Littlewood conjecture $\pi(x+y) \le \pi(x)+\pi(y)$. It seems a bit off, and not only because the conjecture is widely believed ...
3
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0answers
115 views

exponential sum of primes

Fix $\alpha \notin \mathbb{Q}$. I would like to know a reference that shows $$\mathbb{E}_{n\leq N} \Lambda(n) e^{2\pi i \alpha n} \to 0,$$ as $N$ tends to infinity. I am familiar with Vinagradov's ...
5
votes
2answers
378 views

Does the antidiagonal in this square matrix always contain a prime?

Does the antidiagonal in the square matrix with entries $1,2,\ldots,n^2$ row by row in that order always contain a prime? For example: For n=2: $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ ...
3
votes
1answer
147 views

Confusion regarding Riesz Function Definition

According to wikipedia: 'In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series $$ {\rm ...
0
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1answer
157 views

Pruning primitive sequences but still attaining Pillai's lower bound on sum of reciprocals

Background: The answer to a previous question I asked here specified a construction to achieve Pillai's bound on reciprocal sums of primitive sequences. A primitive sequence ...
4
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3answers
527 views

How do estimates on $N_\chi(\alpha,T)$ lead to the Dirichlet prime number theorem for arithmetic sequences?

Let $ N_\chi(\alpha,T)$ be the number of zeros of $L(s=\sigma+it,\chi) = \sum \frac{\chi(n)}{n^s}$ where $c > 0$ and $(\sigma,t) $ are in the rectangle $ [\alpha,1] \times [-T,T]$. In various ...
14
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1answer
552 views

Prove $4\sum_{k=1}^{p-1}\frac{(-1)^k}{k^2}\equiv 3\sum_{k=1}^{p-1}\frac{1}{k^2}\pmod{p^2}$

Wolstenholme's theorem is stated as follows: if $p>3$ is a prime, then \begin{align*} \sum_{k=1}^{p-1}\frac{1}{k}\equiv 0 \pmod{p^2},\\ \sum_{k=1}^{p-1}\frac{1}{k^2} \equiv 0 \pmod{p}. \end{align*} ...
0
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0answers
141 views

The existence of solution for special equation on integer ring

I have a question which belongs to the field of number theory. Can we prove or disprove the following claim: For all prime number $p=24t+1$ and the natural number $n=6t+1$, there is at least, one ...
0
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2answers
449 views

Which even numbers are known to be both prime gaps and the sum of 2 primes?

Goldbach's conjecture asserts that every even integer greater than $3$ is the sum of two primes, while de Polignac's one says every even positive integer is a prime gap infinitely often. My question ...
8
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0answers
123 views

Equation which has nontrivial solutions modulo $N$ for every $N \ge 2$ does not have any nontrivial integer solutions

Let $\alpha = \sqrt[3]{2}$ and $K = \textbf{Q}(\alpha)$. I want to show that the equation$$\text{N}_\textbf{Q}^K\left(x + 4y + z\alpha + w\alpha^2\right) - 6(x + y)\left(x^2 + xy + 7y^2\right) = ...
5
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1answer
457 views

$(n+1)!_\mathbb{P}$ and the Euler-Mascheroni constant

I'm studying the following limit $$\lim_{n\to \infty} \frac{1}{n} \ln\left( \frac{(n+1)!_\mathbb{P}}{n^n}\right) $$ where $$(n+1)!_\mathbb{P} = \prod\limits_{p \in \mathbb{P}}^{} ...
8
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0answers
207 views

Generating prime numbers

By a theorem of Mills, 1947, there is a real number $c$ such that for every $n$, $[c^{3^n}]$ is a prime number. Is there a real number $d$ such that $[d^n]$ is prime, for every $n$ ?
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0answers
108 views

Is there an odd number which has no prime to prime matchings when compared with its reverse order? [closed]

For example look at the number 9. It has prime-prime matching at 3,5, and 7. For example the sequence of 13 has matchings at 1,3,7,11,13. For example 15 has the matchings(crossings) at 3,5,11,13. ...
1
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1answer
182 views

Least simultaneous quadratic non-residue

Suppose $p,q$ are distinct primes with least quadratic non-residues $n_p$ and $n_q$ respectively. Can one bound the least $n$ for which $\left(\frac{n}{p}\right)=\left(\frac{n}{q}\right)=-1$ in terms ...
6
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2answers
262 views

Does the limit of this product over primes converge for all $\Re(s) > \frac12$?

Numerical evidence suggests that: $$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$ with $p_n$ ...
0
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0answers
79 views

Probability distribution associated with total divisors of an integer

Is there a generalization to https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Kac_theorem which gives distribution function for $$\omega(n)=\big|\{d\in\mathsf{prime}:d|n\}\big|$$ where ...
0
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1answer
178 views

When is $a^{2^n}+1$ prime finitely often unconditionally?

Define generalized Fermat numbers following OEIS and mathworld. For natural $a,n$ and $a$ even, the generalized Fermat number (GFN) is $F_n(a)=a^{2^n}+1$. Very large GFN primes are known (in the ...
0
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1answer
128 views

Linear forms that avoid numbers with lot of factors

Is following true? For every given $c>0$ there is an $n_c>0$ such that for every $n>n_c$ there are integers $n<a,b<2n$ such that there are two positive integers $\frac{n}{2(\log ...
4
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1answer
205 views

Goldbach for certain classes of $n$

Asked on MSE without response here. $\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$. The Wiki article on the Goldbach conjecture states that In 1975, ...
2
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1answer
133 views

Gradual monotonic morphing between two natural numbers

Let $a < b$ be two natural numbers. I will use these as an example: \begin{align*} a & = 2^5 \cdot 3^2 \cdot 5^2 = 7200\\\ b & = 2^3 \cdot 3^5 \cdot 7^1 = 13608 \end{align*} I seek to ...
0
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1answer
48 views

Function for number of integers having only prime factors >= x? [closed]

Is there a function for the number of integers below n having only prime factors greater than or equal to p? For example, how do i determine the number of integers below 1000 only having prime ...
3
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2answers
281 views

Primes $P_{2n-1}$ that are $2$ mod $3$

Are infinitely many primes $P_{2n-1}$ expressible as $3k-1$? The primes $P_{2n-1}$ are every other prime beginning with $2$: $2,5,11,17,23,31,\cdots$. The first few are of the form $3k-1$, but $31$ ...
0
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0answers
41 views

Estimates related to sum over a primes from a fixed, possibly sparse set

Let $E$ be a fixed infinite sequence of primes such that $\sum_{p \in E} \frac{1}{p} = \infty$. Assume that $\sigma > 1$ depends on some parameter $x \rightarrow \infty$ in such a way that $\sigma ...
1
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1answer
194 views

Radical of the sum $=$ radical of the product

My question is: Has it been proved/disproved or studied the following? For every $k\geq 4$ there are $k$ pairwise relatively prime numbers $a_1,a_2,...,a_k$ all greater than $1$ such that ...
-5
votes
1answer
232 views

How to prove twin prime conjecture or Goldbach conjecture if we assume prime distribution is completely random? [closed]

If we assume that prime number distribution is COMPLETELY random (subject to 1/log(x) restriction), can we prove twin prime conjecture or Goldbach conjecture ? My feeling is that, this will be ...
5
votes
1answer
267 views

The limit of the following product? What is the closed form of the value?

Assume that $P_n$ is the $n$'th prime: Please help me solve the following $$\lim_{k\to\infty} {k}\prod_{n=1}^k \frac{P_{2n-1}}{P_{2n}}$$ I am not really sure quite where to start here as I am ...
4
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0answers
304 views

A problem on prime numbers

Given integers $a,b,c,d\in[2^n,2^m]$ with $m>n>1$, how many primes $p$ are there in $[n^\alpha,n^\beta]$ for some $1<\alpha<\beta$ such that $$0<a\bmod p<n^{\alpha/k}$$ $$0<b\bmod ...
4
votes
1answer
182 views

What is known about the largest prime divisor of the product of $k$ consecutive integers?

Take $k$ consecutive composite integers from a prime gap. What is known about the largest prime divisor of their product? It seems to me that except for the triplet $(8,9,10)$ and the pair $(8,9)$ , ...
6
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0answers
116 views

Asymptotic density of winning positions in “Prime Nim”?

Consider a single-pile NIM variant, played under standard (not misere) objective, with the rule that you may remove any prime number from the pile. The winning positions of this game are all numbers ...
4
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1answer
310 views

Upper bound for the first Hardy-Littlewood conjecture

About the Hardy-Littlewood conjecture by Terence Tao: Conjecture 2 (Prime tuples conjecture, quantitative form) Let ${k_0 \geq 1}$ be a fixed natural number, and let ${{\mathcal H}}$ be a fixed ...
2
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0answers
324 views

How big can a set of integers be if all pairs have bounded gcd

In this recent MO question, it was shown that the maximal cardinality of a subset $A(M,N)$ of $[1,N]$ where the pairwise GCD's of all set elements are upper bounded by $M,$ with $M^2\leq N$ has size ...
4
votes
1answer
395 views

About factorization in Zhang's proof of weak Twin Prime conjecture

Why does it need to firstly factorize the number n into two factors q and r( Lemma 4 in the paper,see the following)? What's the motivation. What if it doesn't do this factorization?
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2answers
320 views

Does theta(n)<n for all n imply the Riemann Hypothesis and/or vice versa?

I know that better and better bounds of the Chebyshev Theta and Psi functions are implied by knowing that the first (insert large number here) zeta zeroes lie on the Critical Line. These bounds, ...
2
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0answers
143 views

On sets of coprime numbers

We know that from prime number theorem that the number of primes below $n$ and above $\frac n2$ (denoted by $\pi_{n,\frac n2}$ is approximately $$\pi_{n,\frac n2}\approx\frac{n}{2\ln n}.$$ Denote by ...
2
votes
1answer
158 views

Gap between semiprimes

Is there a conjectured gap between semiprimes? There is a conjectured gap between primes in form of Cramer's conjecture. Using this we have $p_1\leq p_0+c(\log p_0)^2$ for consecutive primes $p_0$ ...
22
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1answer
683 views

How big can a set of integers be if all pairs have small gcd?

Suppose $A\subset[1,N]$ is a set of integers. If for any distinct $a,b\in A$ we have $(a,b)\leq M$ then how big can $|A|$ be? If $M=1$ then $|A|$ is at most $\pi(N)$ since the map $a\mapsto P_+(a)$ ...
6
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1answer
188 views

$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$

Assume that $ab \neq 0$. What is $$N_p := \text{card}\{(x, y, z, t) \in (\textbf{F}_p)^4 : ax^4 + by^4 + z^2 + t^2 = 0\}?$$I need this result, but unfortunately I am not a number theorist. Could ...
8
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0answers
168 views

Primes of the form $(2m+1)^2-2^{2s+1}$

The question is the following : Question: Does there exist infinitely many primes of the form $(2m+1)^2-2^{2s+1}$ with $m,s\geq 1$ ? Why this could be true: Bunyakowsky conjecture would ...
2
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0answers
202 views

Avoiding Chinese Remainder Theorem

Given $k\in\Bbb N$ with $k<(\log_2N)^{\frac1\alpha}$ where $\alpha>2$ is fixed and $N$ being some integer such that $$N<\prod_{i=1}^k\pi_i^{a_i}$$ where $\pi_1,\pi_2,\dots,\pi_{k-1},\pi_k$ ...
0
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0answers
109 views

Bounding $\dfrac{r(x)}{\pi(x+r(x))-\pi(x-r(x))}$ with $1\ll r(x)\ll \log^{4}(x)$

I would like to know whether it is possible to obtain the bounds $\sqrt{r(x)}\ll k(x)\ll r(x)$ where $k(x)={\pi(x+r(x))-\pi(x-r(x))}$ and $1\ll r(x)\ll \log^{4}(x)$ and thus ...
2
votes
2answers
344 views

Primes $p$ for which $2p-1$ is prime

It's a well-known open problem (Sophie-Germain primes) whether there are infinitely many primes $p$, $2p+1$. What about $p$, $2p-1$? Seemingly it's also an open problem (see here and the linked ...
3
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1answer
184 views

Is $2^n -1$ finitely many times the product of consecutive primes? [duplicate]

This question was asked at MSE but recieved no attention at all. Here it is: Are there finitely many $(n,k) \in \mathbb{N}^2$ with $2^n-1=p_1p_2\cdots p_k$ ? $p_1=3,p_2=5 , ...,p_k$ are ...
4
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0answers
100 views

Behavior of the “mean prime factor” of numbers

This question concerns the behavior of a function $f(\;)$ that maps each number in $\mathbb{N}$ to its mean prime factor. I previously posted premature questions, now deleted, which explains the cites ...
8
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1answer
332 views

Distribution of the number of prime factors

Count the number of prime factors of a number $n$ to include multiplicity, so that $$n=24=2^3 \cdot 3 = 2 \cdot 2 \cdot 2 \cdot 3$$ has $4$ prime factors, and $$n = 6500 = 2^2 \cdot 5^3 \cdot 13 = 2 ...
3
votes
1answer
281 views

Did Erdős prove there are two primes $4a+1, 4b+3$ between between $n$ and $2n$?

http://mathworld.wolfram.com/ChoquetTheory.html Is the claim in the link true? Here's the reference given there: https://www.renyi.hu/~p_erdos/1934-01.pdf Erdős proved that there exist at least ...
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0answers
124 views

A Question on Chinese Remainder Theorem [closed]

Let $p_1,p_2,\ldots,p_n$ be odd primes. Let $y$ be the unique solution of the Chinese Remainder Problem ( $0 \le y < m$, $ m = p_1\cdot p_2 \cdots p_n$) $ y = \frac{p_1-1}{2} \text{ mod } (p_1) ...
7
votes
3answers
537 views

Quantitative and elementary proofs of the Prime Number Theorem

I would like to know two things: one, whether the best quantative bounds in the Prime Number Theorem are still basically those given by the Vinogradov-Korobov zero-free region? and two, whether there ...
6
votes
0answers
207 views

Any ways to Simplify Daboussi's Argument for Prime Number Theorem?

One strategy to prove the Prime number theorem involves removing some factors: $$ \limsup_{x \to \infty} \underbrace{\frac{1}{x}\sum_{n \leq x} \mu(x)}_{\color{red}{A}}\leq \prod_{p \leq y} \left( 1 ...
4
votes
1answer
487 views

Green-Tao theorem for 1-central numbers

This question came to my mind this afternoon while trying to figure out a possible way to tackle de Polignac's conjecture, which states that every even positive integer can be written as the ...
17
votes
2answers
3k views

Does the equation $241+2^{2s+1}=m^2$ have a solution?

Let $p$ be a prime congruent to $1$ mod. 8. If $p= 17$ one has : $p+ 8 = 5 ^2$. If $p= 41$ one has : $p+ 8 = 7 ^2$. If $p= 73$ one has : $p+ 8 = 9 ^2$. If $p= 89$ one has : $p+ 32 = 11 ^2$. If ...