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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2answers
112 views

About consecutive integers covered by arithmetic progressions [on hold]

Help me please to solve the following problem. There are $n$ arithmetic progressions of the form: $$(2i+1)k + x_i,~~~~ i = 1,\ldots,n, k \geq 0$$ Initial integer terms $x_i \geq 0$ are varying. ...
0
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1answer
182 views

Number of fixed points in Zagier's involution (Fermat's Theorem) [on hold]

Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace $ having ...
2
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1answer
117 views

GCD for two Cullen numbers

The $n$'th Cullen number is $C_n = n\cdot2^n+1$. If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?
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0answers
30 views

Mersenne number with small Carmichael function

Let $\lambda(\cdot)$ be the Carmichael function. I'm trying to understand the magnitude of the smallest values of $\lambda(2^n - 1)$, when $n$ runs over the positive integers. Precisely my question is:...
32
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2answers
2k views

What did Yu Jianchun discover about Carmichael numbers?

There's a news story going around (see for example [1]; other accounts are even more breathless) about an amateur mathematician, Yu Jianchun, finding an "alternative method to verify Carmichael ...
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1answer
157 views

When the Ratio of Two Factors is a power of $2$ (i.e. $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$)

We can write, $n!= 2^s \times a \times b$ where $gcd(a,b)=1$ and $2^{s+1} \nmid n!$. Problem: Is there infinite number of $n$ when $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$? Question: 1.How ...
2
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1answer
109 views

An upper bound for the number of prime numbers in non-linear progressions

Let $f(x)$ be a non-linear polynomial over $\mathbb{Z}$. Consider the following sum $\pi_f(x):= \#\{y: y< x \text{ and } f(y) \text{ is prime} \}$. Can we get an upper bound for $\pi_f(x)$?
2
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0answers
66 views

The largest primes in the monster group construction

The last three prime numbers in the factorization of the order of the Fischer Griess friendly monster group are 47, 59, 71. (https://en.wikipedia.org/wiki/Monster_group) On the other hand, the monster ...
0
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1answer
171 views

Prime quadratic non-residue

NC Ankeny showed assuming Riemann Hypothesis the least quadratic non residue( let it be '$r$') modulo some prime $p$ to be $O(\log^2 p)$. It is easy to see that $r$ is a prime. I have following ...
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0answers
152 views

Deduction formula for Goldbach counting function

Assume $N\geq 1$ is integer and $P\geq 1$ is square-free integer. Goldbach counting function, $S_P(N,x)$, is defined to be the number of $n$ between 1 and $x$ such that $(N-n)(N+n)$ is co-prime to $P$....
2
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2answers
270 views

Asymptotic estimate of the probability of $(n, P(\sqrt{x})) \leq x$?

Let $P(x)$ be the product of all primes less or equal to $x$. The probability of $(n, P(\sqrt{x})) \leq x$ for an arbitrary $n$ is then given exactly by $$ \prod_{p\mid P(\sqrt{x})}{\left(1-\frac{1}{p}...
1
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1answer
160 views

Primes and the factorial

Conjectures: For all prime numbers $p>5$ there exist a prime number $q<p$ such that $q\equiv m!\!\pmod p$, for some $m$ with $2<m<p$. and For all primes $p$ there is a prime $...
1
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2answers
177 views

Primes p(n) such that p(n) + p(n+3) = p(n+1) + p(n+2) and p(n) + p(n+4) = p(n+2) + p(n+3) - Conjecture [closed]

I have been studying this sequence (A266882 in the OEIS) and found the following pattern: $13 + 17 + 19 + 23 + 29 = 101$ (101 is prime) 37 does not hold. $223 + 227 + 229 + 233 + 239 = 1151$ (1151 ...
0
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1answer
141 views

Minimal number of different values in the sequence $(\mu(d_i)\varphi(d_i))_{i=\overline{{1,\tau(m)}}}$

Let $m=p_1\ldots p_k$ be the prime factorization of some positive integer $m$ and $k\geq 2$. Let $d_1,\ldots,d_{\tau(m)}$ be all divisors of $m$, where $\tau(m)$ counts the number of divisors of $m$....
0
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0answers
147 views

How do these primes jump?

I have several questions regarding the analysis, behaviour, and expression of a simple sieving algorithm which uses associative arrays. The pseudocode below assumes integer addition, string ...
1
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2answers
632 views

Statements going against the grain of Riemann Hypothesis (R.H.) [closed]

Let $M(N) := \sum_{n=1}^N \mu(n)$ It is known that bounding $M(N)$ by $N^{1/2+\epsilon}$ implies R.H. A bound of $M(N)$ by $K\sqrt N$ for say $ K\ge2 $ is also sufficient. $K = 1$ is excluded as ...
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0answers
78 views

A rash guess about distribution of primes based on meager empirical evidence?

Elementary number theory is a field in which imbeciles can ask questions that experts cannot answer (and I wonder if discrete geometry is a similar subject in that respect?) and herewith I submit ...
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1answer
135 views

product distinct prime factors of prime(n)-1 and prime(n)+1

The prime 127 has 127-1=126 with distinct prime factors 2,3,7 and 127+1=128 with distinct prime factors of only 2; hence 2*3*7=42<127. Log 127/42=q=1.296. Are such primes common? Can a value of ...
13
votes
1answer
396 views

Does Littlewood's bound on $\zeta(1+it)$ extend to all the partial sums?

Littlewood established that $2e^{\gamma} \geq \limsup_{t \to \infty} |\zeta(1+it)| / \log{\log{t}} \geq e^{\gamma}$, the lower bound unconditionally and the upper bound on RH. It now seems to be ...
1
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0answers
86 views

Reference request: Bounding exponential sum $\sum_{x \in [0,X]} \Lambda(x) e(\beta_d x^d + \ldots + \beta_1 x )$

Let $1 \leq i \leq d$, $q \in \mathbb{N}$, and $0 \leq a_{i} < q$. Let $$ \mathfrak{M}^{(i)}_{a_{ i}, q} (C) =\{ \beta_{i} \in [0,1) : | \beta_{i} - a_{i}/q | \leq (\log X)^{C} X^{-i} \} . $$ We ...
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1answer
112 views

Is a positive integer determined by its sequence of typical primality radii?

This question is a follow-up to About Goldbach's conjecture . Assuming the truth of Goldbach's conjecture, suppose $n$ and $m$ are two positive integers such that $N_{2}(n)=N_{2}(m)=:N$ and that ...
5
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1answer
257 views

Infinitely many primes coming from Euclid's proof

When teaching Euclid's classic proof of the infinitude of primes today, the following question appeared to me. Let $p_1,p_2,p_3,\ldots$ be the prime numbers, listed in increasing order. Set $$k_n = ...
3
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2answers
156 views

$p$-simple integers from between $n$ and $n+p-1$

Let $\ p\ $ be an arbitrary prime. Then an integer $\ s\ $ is called $p$-simple $\ \Leftarrow:\Rightarrow\ s\ $ is not divisible by any prime $\ q<p.\ $ Could you prove my conjecture (or is it ...
9
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0answers
278 views

Is the number $\sum_{p\text{ prime}}p^{-2}$ known to be irrational?

Is the number $$\sum_{p\text{ prime}}p^{-2}$$ known to be irrational? The limit exists, since $$\sum_{p\text{ prime}}p^{-2}<\sum_{i=1}^{\infty}i^{-2}=\frac{\pi^{2}}{6}$$.
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0answers
86 views

Recursive construction of integers from Fermat Primes

Assume there are only finitely many Fermat primes. Fix a Fermat prime $F_t$ say $3$. What is in general known about numbers of form $F_{t,k}=2^kF_t +1$? Is the number of primes of form $F_{t,k}$ ...
9
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0answers
195 views

Greedy permutation of the set $\{1,2,\dots,n\}$ and prime numbers

Answering this question, another question came to my mind. For which $n$ will the greedy algorithm work? We define a sequence of natural numbers $x_n$ recursively: $$x_1 =1,$$ $$x_n \mbox{ is the ...
7
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1answer
263 views

Permutations of the set $\{1,2,…,n\}$ and prime numbers

Here is the version of this question that I posted on math.stackexchange a few days ago and I did not receive an answer that settles my question so I thought that maybe on this site I could get a ...
7
votes
1answer
181 views

Main term in the number of sign changes of $\psi(x) - x$

Define $N_\Delta(T)$ to be the number of sign changes of $\psi(x) - x$ in the interval $[1, T]$. Landau's Theorem says $N_\Delta(T)$ is $\Omega(\log T)$ [1]. But perhaps that estimate is too crude. ...
1
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1answer
475 views

What is the best currently proven bounds on prime gaps?

I did some digging around on the internet but I found tons of different equations on both lower and upper bounds for the largest possible prime gap g(n). I was wondering what are currently the best ...
0
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0answers
42 views

On a count of certain number of primes in an interval

Fix a prime $p$, $\alpha\in(0,1)$ and $\beta\in(1,2)$ and let $\mathcal U$ be primes in $[p^\alpha,\beta p^\alpha]$ such that if $b\in\mathcal U$ and if $d$ is multiplicative inverse of $b$ in $\Bbb ...
1
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1answer
244 views

Density of Sophie Germain $3\bmod 4$ primes

Is there any reason to expect the density of primes $p$ such that $2p+1$ is also a prime where $p=3\bmod4$ holds would be different from case of $p=1\bmod4$? What if $2p+1$ is replaced by $2p-1$ and ...
7
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1answer
152 views

Density of prime divisors of $a^n + b$

Suppose $a > 1, b \neq 0$ be two rational numbers. Is it known in general that the set of prime divisors of (the numerator of) $a^n + b$ has a positive relative density?
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0answers
178 views

Effective prime number theorem

The prime number theorem implies that for every $ϵ>0$, there is $n_\epsilon$ such that for all $n≥n_\epsilon$ the number of primes in $[n,cn]$ is at least $\frac{(c−1−\epsilon)n}{\log n}$ and at ...
0
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0answers
48 views

Density of set of primes which avoid given finite set of residues modulo powers of all primes

Let $k\in\mathbb{N}$, $k\ge2$ and $S\subseteq\mathbb{Z}$ be a finite set of integers. For every prime $p$ let $c_p$ be a number of invertible residue classes mod $p^k$ that contain some element of $S$....
4
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0answers
283 views

Proof for new deterministic primality test possible?

Conjecture: Let $n \in \mathbb{N}$ and $n$ odd. Then the number $N=2^2 + n^2$ is prime, if and only if $N$ divides $2^{(N-1)/2} + 1$. Thanks.
7
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1answer
164 views

Do arbitrary $K$-twin primes exist?

Polignac's conjecture states that for any positive even integer $K$, there exist infinitely many pairs of primes such that their difference is $K$. I am interested the status in a much weaker form of ...
2
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3answers
352 views

Primes in arithmetic progressions in number fields

My general question is how does one prove equi-distribution results for primes in arithmetic progressions in number fields? I am interested in the equi-distribution of prime elements of the ring of ...
2
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0answers
117 views

$f(x)$-th largest number of prime factors

Given a sufficiently well-behaved function $1\le f(x)\le x$ and a multiset $S=\{\omega(n): 1\le n\le x\}$, what can be said about the asymptotics of the $f(x)$-th largest member of $S$? In other words,...
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0answers
136 views

On the cardinality of the set of right-truncatable primes

We say that the (base ten) prime number $p=a_{n}a_{n-1}a_{n-2}\cdots a_{1}a_{0}$ is right-truncatable if all of the following numbers are prime: \begin{eqnarray*}a_{n},\\a_{n}a_{n-1},\\ a_{n}a_{n-1}...
8
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1answer
241 views

Arbitrarily many primes in a Fibonacci-type sequence

It is conjectured that the standard Fibonacci sequence contains infinitely many primes. While this is perhaps too difficult, I am wondering about the following simpler version: Question. For any $K$, ...
3
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1answer
437 views

origin of analogy “primes as the atoms of number theory/ arithmetic”

a math student recently challenged me on the old comparison/ analogy of prime numbers to "the atoms of number theory or arithmetic" and then was wondering the origin of the phrase. where does this ...
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0answers
166 views

Asymptotic value of sum over Möbius function

Consider the sum $$ S(x)=\sum_{\substack{{d\mid P(\sqrt{x})}\\{d\leq x}}}|\mu(d)|, $$ where $P(x)$ is the product of all primes less than or equal to $x$, and $\mu(d)$ is the Möbius function. Q:...
3
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1answer
118 views

Reference Request on the existence of $k$ satisfying $P(\Phi_d(2))^k \gt \Phi_d(2)$ for all $d$

I am working my way through the literature regarding the following conjecture: There is a positive integer $k$ such that for all positive integers $d$, $$P(\Phi_d(2))^k \gt \Phi_d(2).$$ I am ...
20
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1answer
1k views

How many primes can there be in a short interval?

Given $n \in \mathbb{N}$, let $\pi(n)$ denote the number of prime numbers $\leq n$. What is $$ \limsup_{m \rightarrow \infty} \left( \limsup_{n \rightarrow \infty} \frac{\pi(n+m) - \pi(n)}{\pi(m)} \...
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1answer
130 views

Find the values of $n$ that satisfy this inequality involving a product over prime numbers [closed]

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the $i$th ...
4
votes
1answer
245 views

The values of $n$ which satisfy an inequality about prime numbers

For which values of $n$ does the following inequality hold for? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the $i$-...
2
votes
1answer
89 views

Set of triple-primes satisfying a certain equation

Is there a set of triple-primes satisfying the following equation? $p_1p_2+p_2p_3+p_3p_1+p_1+p_2+p_3=2^β,p_1p_2p_3=2^α-1,α>β.$ I have checked the first 11 numbers that no one satisfy the above ...
18
votes
1answer
314 views

On random divisor sums modulo $2^k$

Let $k,n,\ell$ be positive integers with $k,n\ge 2$ and $0\le \ell \le k-1$. For each integer $2\le j \le n$, choose a divisor $d_j$ of $j$, uniformly at random from the divisors of $j$. We denote by $...
7
votes
0answers
140 views

k-Almost Primes in short intervals

According to this question every interval $[x, x + x^{0.45}]$ contains a product of two primes, and this has been improved further slightly. Are there better results available for $k$-almost primes? ...
6
votes
1answer
233 views

Can $b^4+1$ be a pseudoprime to base 2 (except for Fermat numbers)?

Cross-post: This very elementary question was first posted to Mathematics Stack Exchange but the response I got there (even after offering a bounty) was not useful. For the purpose of this question, ...