Questions tagged [prime-numbers]

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 specialized tag in combination with the top-level tag nt.number-theory and (if applicable) analytic-number-theory.

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Primality test for numbers of the form $\frac{a^p-1}{a-1}$?

Here is what I observed : Inspired by Lucas-Lehmer primality test, I think a made a primality test for numbers of the form $\frac{a^p-1}{a-1}$ but the test isn't perfect and there are some conditions ...
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-4 votes
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42 views

Closest multiples of 2 prime numbers [closed]

How can we find the distance between the closest multiples of 2 prime numbers (let's say a and b) smaller than a x b?
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3 votes
1 answer
107 views

Primality test for $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ and $\frac{(10 \cdot 2^n)^m+1}{10 \cdot 2^n+1} - 2$

Here is what I observed: For $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ : Let $N$ = $\frac{(10 \cdot 2^n)^m-1}{10 \cdot 2^n-1} - 2$ : when $m$ is a number $m \ge 3$ and $n \ge 0$. Let the ...
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3 votes
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129 views

What can be said about the primality of Zsigmondy numbers?

I am cross-posting this from math.stackexchange, as it has received upvotes but no comments/answers after a couple months. Let $\mathcal{Z}(n,a,b)=\frac{\Phi_n(a,b)}{\gcd (\Phi_n(a,b),n)}$ be the $n$-...
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1 vote
0 answers
79 views

A definition related to pseudoprimes and the Dedekind psi function

In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
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5 votes
2 answers
747 views

A question regarding Cramér's proof on prime gaps under the Riemann Hypothesis

Let $p_n$ be the $n$th prime. Assuming the Riemann hypothesis, Harald Cramér proves that $p_n-p_{n-1}\le C(\sqrt p_n \log p_n)$ for sufficiently large $n$. Is there a value known for the constant $C$ ...
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1 vote
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Prime numbers formed by consecutive numbers

Let $x$ be an even number. Suppose that we concatenate it with its successor to form $x (x+1)$ (not multiplication, but concatenation). For example, if we start with $x = 2$, we would get $23$. If we ...
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  • 411
14 votes
2 answers
1k views

Sum of reciprocals of Sophie Germain primes

A prime $p$ is called a Sophie Germain prime if $2p+1$ is also prime: OEIS A005384. Whether there are an infinite number of such primes is unsolved. My question is: If there are an infinite number of ...
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2 votes
0 answers
233 views

How soon can we represent a number as the sum of two primes?

Posting in MO since it was unanswered in MSE. Goldbach's conjecture says that every even number can be represented as the sum of two primes. But how soon can we find such a representation. Taking $20 =...
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111 views

On general 'explicit' expression for constant term in finite sum of function of primes

Consider the following finite sum $$\sum_{p\leq x}f(p) = S(x)+C$$ Here, $f(x)$ is smooth $p$ is prime $S(x)$(=smooth+oscillation) is also a 'function'; $C$ is a constant We also know the following $$\...
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2 votes
1 answer
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Reference request: Numbers composed of given primes

The number of $n \leq x$ composed using only the given primes $p_1, p_2, ... p_k$ as $x \rightarrow \infty$ satisfies $$ \frac {\log^k x} {k! \prod_1^k p_j} + O \left ( \log^{k-1} x \right ) . $$ I am ...
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3 votes
1 answer
365 views

Root of polynomials in a finite field

I am looking for a way to find out if a polynomial $P\in \mathbb Z/p\mathbb Z=\mathbb F_p$, of great degree, has roots in $\mathbb F_p$, with $p$ a big prime number. For example : $p=2^{2020}-69$ ...
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1 vote
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Is there a known connection between Wieferich primes and the Goormaghtigh conjecture?

I posted this question on SE, and was told I should repost it here. The Goormaghtigh conjecture explores the Diophantine equation of the form $$ \frac{a^b-1}{a-1}=\frac{c^d-1}{c-1}, $$ where $a>c&...
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5 votes
1 answer
126 views

Primes in arithmetic progressions: weak version of Linnik's theorem with prime power modulus?

Looking at a problem in representation theory I ran into a question on small primes in arithmetic progressions. Let me begin with a short summary of results on small primes in arithmetic progressions. ...
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3 votes
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304 views

Proof of an explicit formula for $\pi_0(x)$

Let $\pi(x)$ denote the prime counting function and $$\pi_0(x) = \lim_{\epsilon \to 0} \frac{\pi(x+\epsilon)+\pi(x-\epsilon)}{2}.$$ I've seen noted in a few references the explicit formula $$\pi_0(x) =...
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1 vote
1 answer
159 views

Counting cubic residues mod p

Given a prime $p=3m+1$, $(p-1)/3$ of the residues mod $p$ are cubic residues. So heuristically, for any given integer $k>1$ not a perfect cube, we would expect that about 1/3 of the primes $\equiv1\...
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1 vote
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Are there any connections between $a$ and $c$ where $p = a^2 + 2b^2 = c^2 + d^2$?

Let $p$ be a prime such that $p \equiv 1 \mod 8$. Then we know there exists $a,b \in \mathbb{Z}$ such that $p = a^2 + 2b^2$. But at the same time $p \equiv 1 \mod 4$, so there also exists $c,d \in \...
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140 views

Primality of $\ \frac{b^p-a^p}{b-a}$

Can you prove the following: Conjecture:  Let $\ p\in\mathbb P\ $ be an arbitrary prime. Then there exist two relatively prime integers $\ a\ $ and $\ b\ $ such that $\ a>0\ $ and $\ b>1\ $ and $...
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7 votes
2 answers
648 views

Estimates about prime numbers: a lemma in Bourgain's article

For $n\in \mathbb{N}$ with prime decomposition $n=p_1^{r_1}\cdots p_k^{r_k},p_i\neq p_j$, let $A=\{p_1,\cdots,p_k\}$; then the following holds: \begin{equation} |\{q\in \mathbb{N},q<Q: \text{all ...
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5 votes
0 answers
144 views

Proofs in number theory that involve non-standard models of arithmetic

While reading an introductory text on model theory, I found it interesting that one can reformulate the famous conjectures about twin primes and Mersenne primes in terms of non-standard models of ...
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7 votes
2 answers
531 views

Solving functional equation $f(xy)=f(x+y)$ and Diophantine equations

It is well known that the only solution is $f$ a constant function. However, by putting some restrictions on the functional equation, we might get other solutions, with potential implications to ...
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1 vote
0 answers
208 views

Liu's new sieve weight

Does Liu's sieve weight (in his arXiv paper "On the gap between primes") $$sieve(n)=(\sum_{\substack{d_i\mid (n-h_i),i=1,\cdots,k\\ (d_1,\cdots,d_k)\in\mathcal{D}}}\lambda_{d_1,\cdots,d_k} ...
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1 vote
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Can the prime gap record of Liu be improved further?

Let $d$ be the least positive integer such that there are infinitely many distinct prime pairs $\{p,q\}$ with $|q-p|\le d$. The twin prime conjecture is equivalent to $d=2$. In 2013 Yitang Zhang ...
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65 votes
2 answers
3k views

Function that produces primes

For any $n\geq 2$ consider the recursion \begin{align*} a(0,n)&=n;\\ a(m,n)&=a(m-1,n)+\operatorname{gcd}(a(m-1,n),n-m),\qquad m\geq 1. \end{align*} I conjecture that $a(n-1,n)$ is always ...
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0 votes
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176 views

Does Cramer's random model of primes imply $L(n)w_{0}(n)=O(\log^{4}n)$?

Say $r$ is a Galois radius of $n$ of level $l=ab$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime. Denote by $r_{l,0}(n)$ the smallest non negative Galois radius of $n$ of level $l$, by $\...
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58 views

Duality between primality radius and Galois radius of maximal prime level

Say $r$ is a Galois radius of level $l=ab$ of $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime, and let $\rho:=\frac{r}{l}$ the corresponding normalized Galois radius of $n$. Denote by $r_{l,k}(n)...
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2 votes
0 answers
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Is the inequality $\frac{2r_{l,0}(n)}{K_{d,0}(n)}\lesssim\log^{a+b}n$ provable for some values of $a$, $b$ and $d$?

Say $r$ is a Galois radius of level $l=ab$ and of type $(a,b)$ of $n$ if $n-r=p^a$ and $n+r=q^b$ with $p$ and $q$ prime. Let $r_{l,0}(n)$ the smallest non negative Galois radius of $n$ of level $l$ ...
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76 views

Consecutive integers with consecutive primality radii

Disclaimer: the following observation is purely empirical and as such may not suit this website. Still, it may reveal interesting patterns about primes so I decided to ask this question nevertheless. ...
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2 votes
0 answers
131 views

A conjecture about prime test

Conjecture If $\varphi(m)<\varphi(n)$ for all $m<n$,then $n$ is a prime number. I tried to find a counterexample when $n=pq$ ($p,q$ are prime), then we have to find a prime between $(p-1)(q-1)$ ...
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3 votes
1 answer
154 views

A new convolution, on function of $\mathbb F_p^n$ to $\mathbb F_p$ still zero?

Let $p$ integer prime, $f$ a function of $A=\mathbb F_p^n$ to $\mathbb F_p$, with $n\geq p+1$. Is it true that : for all $x\in A, \sum\limits_{\sigma \in S_n} s(\sigma) \times f(x_\sigma) =0$? $s$ ...
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3 votes
1 answer
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Numbers $n$ whose representation as the product of two divisors require more digits than that of $n$

Note: Posting in MO since it was unanswered in MSE Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > ...
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0 votes
1 answer
131 views

Nomenclature for largest odd factor

Is there a standard phrase for the largest odd factor of a positive integer $n$, or more generally for $n$ divided by the largest power of $p$ that divides it (with $p$ some fixed prime)? Five minutes ...
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68 views

$\max\left\lbrace(n-i+1)\operatorname{prime}(i); 1 \leqslant i \leqslant n\right\rbrace$ from permutation

Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and sorting in descending order if $n$ is prime sorting in ascending order if $n$ is not prime every $n$ ...
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4 votes
1 answer
253 views

Parities of binary weights of primes

Let $X$ denote the sequence A200246: an $i$-th element of $X$ is equal to $w(p_i) \bmod 2$, where $w(p_i)$ is the number of ones in the base-$2$ representation of an $i$-th prime. The first $564163$ ...
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0 votes
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153 views

A question on (trigonometric) prime counting function and twin prime counting function

(I myself don't think the following question is suitable for this forum but as it contains something related to twin primes, I asked here.Please help accordingly) Consider the following sum: $$S(t)=\...
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3 votes
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191 views

No perfect patterns in the primes

The primes are equidistributed in the residue classes $1(\!\!\!\mod{4})$ and $3(\!\!\!\mod 4)$. We also know (for example, by Rubinstein-Sarnak) that the patterns cannot be eventually alternating, i.e....
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8 votes
3 answers
520 views

Why is there an unexpected increase in the density of certain types of Goldbach primes?

Note: Posted in MO since it was unanswered in MSE. I was checking how quickly we can verify Goldbach's conjecture for a given even number $n$ and it was clear that searching backward starting from the ...
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2 votes
0 answers
104 views

Error term related to partial twin prime constant

The twin prime constant is defined as $$ \Pi_2 = \prod_{\substack{p\in\mathcal{P}\\p>2}}\frac{p-2}{p-1}\left(1-\frac{1}{p}\right)^{-1}, $$ where $\mathcal{P}$ is the set of primes. I'm interested ...
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3 votes
2 answers
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What is the importance of Polignac’s conjecture?

The twin-prime conjecture (also known as Polignac’s conjecture, 1846) states that there are infinitely many twin primes (pairs of primes that differ by 2; for example, 3 and 5, 5 and 7, 11 and 13, and ...
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3 votes
2 answers
283 views

How to explain a particular property of the second-to-last bits of primes?

Assuming that $i \geq 0$, let $p_i$ denote an $i$-th prime: $p_0 = 2, p_1 = 3, \ldots$ Then $b_i$ denotes the second-to-last bit of $p_i$, i.e. $b_i = \left\lfloor p_i/2 \right\rfloor \bmod 2.$ The ...
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0 votes
0 answers
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Prime numbers from another permutation

Related question: Prime numbers from permutation Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n-1)$ fixed and forward-cyclically permuting every $n$ consecutive ...
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14 votes
1 answer
283 views

Unpublished result of Rosser in Sieve Methods book

Erdős and Selfridge (1971) state that the following is "implied by an unpublished result of Rosser" which they claim appears in a forthcoming book on sieve methods by Halberstam and Richert. ...
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5 votes
0 answers
103 views

Residues of consecutive primes modulo a fixed integer

It is well-known that the primes are uniformly distributed in residue classes modulo any fixed integer. More precisely, for each integer $q$ and each residue $a \in \mathbb{Z}/q\mathbb{Z}$ that is ...
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9 votes
1 answer
482 views

Prime numbers from permutation

Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and reverse-cyclically permuting every $n$ consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $...
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1 vote
0 answers
69 views

Jones–Sato–Wada–Wiens diophantine equation [closed]

I came across this from the 1993 book Matiyasevic - Hilbert's 10th problem. Typeset from another question: \begin{align} P(a,b,\dotsc,z)=(k+2)\Bigl(1&-(wz+h+j-q)^2\\ &-\left[(gk+2g+k+1)...
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8 votes
3 answers
660 views

Is there some example that nicely extends the multiplication of natural numbers?

Motivation: In mathematics, it is natural to decompose a complicated thing into simpler ones. In the system of natural numbers, the process to decompose large number is to factor it. The ...
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6 votes
0 answers
445 views

Existence of an explosive prime

The motivation to introduce explosive prime is Carmichael's totient conjecture (see why below). Let $\mathbb{N}_{SF}$ be the set of positive square-free integers. Consider the map $f:\mathbb{N}_{SF} \...
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4 votes
1 answer
199 views

Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a prime number

Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it ...
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3 votes
1 answer
134 views

Zsigmondy's Theorem Generalization

Zsigmondy's Theorem states that if $a>b>0$ are coprime integers then for any integer $n\geq 1$ there is a prime $p$ that divides $a^n-b^n$ and does not divide $a^k-b^k$ for any positive integer $...
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1 vote
0 answers
67 views

Symmetric Prime Tuples in arithmetic progressions [closed]

Some time ago I made a post How to make a pair of six-sided dice whose sum is always a prime number? on Math.StackExchange. Now that I'm finishing my studies I decided to approach the problem again ...
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