Questions tagged [prime-numbers]

Prime numbers are natural numbers greater than 1 not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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To solve for $x,y,n$ in non-negative integers , $\dfrac{x!+y!}{n!}=p^n$ , $p$ a given prime

Let $p$ be a given prime , then how do we find non-negative integers $(x,y,n)$ $\space$ , such that $\dfrac{x!+y!}{n!}=p^n$ ?
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How to list the prime factorised natural numbers?

Today I set out to invent a two character numeral system designed to make factorization trivial. Indeed, it lets one factor non-trivial numbers with over thousand digits within 30 seconds per hand - ...
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Are [Wieferich] primes the only solutions to $2^{n-1} \equiv 1 \pmod{n^2}$?

While studying a certain Diophantine equation in the integer $k \ge 2$, I believe I have proven the necessary restriction $$2^{k-1} \equiv 1\!\!\pmod{k^2}. \qquad(\star)$$ Based on what I read ...
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Coprime multiplicative orders modulo infinitely many primes

Is it true that there are infinitely many primes $p$ such that the multiplicative orders of $2$ and $3$ are coprime $\pmod{p}$? By this I mean their order in $(\mathbb{Z}/p\mathbb{Z})^*$. If the ...
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Statement about Woodall primes.

A Woodall number is an integer of the form $n 2^{n}-1$. A Woodall prime is an integer that is both a prime and a Woodall number. Let $p$ be a prime of the form 1 mod 4. Then $p 2^{p} -1$ is never a ...
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Partial summation of a harmonic prime square series (Prime zeta functions)

I am trying to find the following series: $S=\displaystyle\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\dfrac{1}{p_ip_j},A\leq p_1 < p_2 < \dots < p_n \leq B, \lbrace A,B\rbrace \in \mathbb{N}$ ...
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Density of products of a certain set of primes

I have a set S of prime numbers and I would like to find the size (in some sense, ideally some nice asymptotic expression) of the set of positive integers which are the product of with all prime ...
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Number of primefactors in $ f(n,W) = \prod_{k=1}^W (p_k^n -1) \text{ where } p_k=Prime(k) $

I'm reviving an old fiddling, although I do not yet really see its benefit. Beginning with the eulerproduct for the zeta-function in the representation $\small \zeta(n)=\prod {1\over 1-p^{-n} } = \...
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From prime to prime by squaring the digits

I took prime $131$, squared digits of it and wrote them in natural order as they appear, from left to right, and obtained $191$, then I obtained $1811$ by the same procedure, and then $16411$ and then ...
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Conjectures on the primality of $\sum\limits_{i=1}^n p_i^{p_i}$ and $\sum\limits_{i=1}^n (-1)^ip_i^{p_i}$

Inspired by Is $29$ the only prime of the form $p^p+2$?. Claims on prime powers and their alternating sums Consider the expressions $\mathcal P=\sum\limits_{i=1}^n p_i^{p_i}$ and $\mathcal Q=\sum\...
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The quadrature of the circle: comparing Archimedean and Ulam spirals

There are two closely related arrangements of the natural numbers that allow to show patterns in the distribution of some sets of numbers (multiples of 2, 4, 8, square numbers, prime numbers): the ...
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Almost a prime number recurrence relation

For the number of partitions of n into prime parts $a(n)$ it holds $$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$ where $q(n)$ the sum of all different prime factors of $n$. Due to https://oeis....
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Conjecture: $n>2$ is prime iff $\sum_{k=1}^{n-1}\left(3^k-2\right)^{n-1} \;\equiv\; n \cdot 2^{n-1}-1 \pmod{\frac{3^n-1}{2}}$

This question is closely related to: Conjectured primality test Can you provide a proof or a counterexample for the following claim : Conjecture. Let $n$ be a natural number greater than $2$. Then ...
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Prime number intercept

Suppose I arrange my (infinite) list of prime numbers in the following way: \begin{array}{c|c}x_i&2&5&11&17&23&31&\cdots\\\hline y_i&3&7&13&19&29&37&...
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Questions related to the Dirichlet series for $\frac{\zeta'(s)}{\zeta(s)^2}$

This question is related to the following two functions evaluated with the coefficient function $a(n)=\mu(n)\log(n)$. (1) $\quad f(x)=\sum\limits_{n=1}^x a(n)$ (2) $\quad\frac{\zeta'(s)}{\zeta(s)^2}=...
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Product of sum of reciprocals

For any positive integers $k$ and $l$, does the equation $$(\sum_{i=1}^k \frac{1}{p_i}) (\sum_{j=1}^l \frac{1}{q_j}) = 1$$ have solutions in distinct primes, that is, $p_1, p_2, \dots, p_k, q_1, q_2, \...
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Primitive Trinomial for $82589933$?

At Twelve new primitive binary trinomials, $x^{74207281}+x^{9999621}+1$ is shown to be a primitive trinomial in $GF(2)[x]$. Note that $2^{74207281}-1$ was the largest known (Mersenne) prime before ...
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Primes of the form $p=x^4+y^4$

Are there infinitely many prime numbers $p$ such that $$p = x^4+y^4$$ for some $x,y \in \Bbb Z$ ? What if we only require $x,y \in \Bbb Q$ ? I know that $p = a^2+b^2$ with $a,b \in \Bbb Q$ iff $p = a^...
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Prime number theorem for knots?

Is there an analog to the prime-number theorem describing the distribution of the prime numbers among the integers: A theorem that describes the distribution of the prime knots, perhaps with respect ...
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Disproving the existence of a specific infinite sequence of Fibonacci primes

Consider the following sequence: $$ T_{1} = a,\: T_{i+1} = F_{T_{i}} $$ where $ a \in \mathbb{P} $ and $F_i$ is the $i$-th Fibonacci number. Is there a value of $a \neq 5$ such that this sequence ...
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Is $n=13$ the only solution to this: $\pi\left(\sum_{i=1}^n\pi(i)\right)-1=n$.

I was messing around with the prime counting function $\pi(n)$ because I was bored, but then I noticed something. The equation $$\pi\left(\sum_{i=1}^{n}\pi(i)\right)-1=n,$$ has the ...
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Conjecture on product of first $n$ primes.

I was recently studying about reduced residue systems , and then I stumbled upon an idea. Let $P$ represent the set of all primes. Let $P_{n}\#$ denote the product of the first $n$ primes and let $...
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Ratios of prime gaps $(p_{n+1}-p_n)/(p_{2n+1}-p_{2n})$

This is a question about prime gaps $g_n = p_{n+1}-p_n$ that started with a look at the average of ratios $$r_n=\frac{p_{n+1}-p_n}{p_{2n+1}-p_{2n}}$$ and of the inverse, $$ s_n=\frac{p_{2n+1}-p_{2n}}{...
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Is there another prime $p$ such that $S(p)$ is prime?

Denote $$S(p):=2^2+3^3+5^5+\cdots +p^p$$ $S(p)$ is prime for $p=3,7,89$. Is there another prime $p$ such that $S(p)$ is prime ? Is the number of primes $p$ such that $S(p)$ is prime, finite ?
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Prove that there exist infintely many primes $p$ with $p\equiv2$ or $-2 \pmod{5}$

I want to show that there exist infintely many primes $p$ with $p\equiv2$ or $-2 \pmod{5}$. I already know a few proofs for primes with given modulo, but I don't know how I would use that here. ...
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Proof that every even integer $>2$ can be expressed as the sum of two times some prime $p$ and the distance between $p$ and a greater or equal prime?

I'd like to know if there is some proof on this. Suppose we have an even integer $n$, then the conjecture tells us that you can express it in the form of $n=2p+g$, where $p$ and $g$ are some prime ...
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Finding a Formula for a Sum

Given integers $1\le j\le n$, let $p$ denote the largest prime at most $n$. I want to sum $$1/i$$ over all $i=2^{a_2}3^{a_3}\cdots p^{a_p}$ $\,(a_l\ge 0)$ such that both $j,n$ have at least 2 more ...
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Is my proof that there are infinite primes actually valid?

I was trying to think of another way of showing that there are an infinite number of primes. I came up with the following argument, but I am not sure if it is valid. I don't know how to make it more ...
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Primes and square offsets

For a prime $p$, define $\delta(p)$ to be the offset $d$, smallest in absolute value, such that $p \pm d = r^2$ for $d,r \in \mathbb{N}$. For example, \begin{eqnarray} \delta(11) & = & -2 \;:\...
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The number of distinct least prime factors in a sequence of consecutive integers

I was thinking about the number of distinct least prime factors in a sequence of consecutive integers and I noticed: That every $4$ integers, there are $3$ distinct least prime factors. Every $6$ ...
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Fibonaccis and prime numbers

Let $F_n$ denote the $n$th Fibonacci number, and $p_n$ the $n$th prime. Let $a(n)$ be the smallest positive integer such that $p_n$ is a factor of $F_{a(n)}$. How can I see that it follows that ...
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Four primes together in an equality. How often does it happen?

I am almost a complete number theory newbie so pardon me if this is stupid. $$128 - 125 = 3$$ $$2^7 - 5^3 = 3$$ Are there an infinite number of these, four primes $p_1,p_2,p_3,p_4$ so that: $${p_1}^...
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Help finding the flaw in this proof

Today I spent my afternoon trying to understand why the Hardy-Littlewood's Second Conjecture is considered as a problem of such a great difficulty. I got a fairly strange result, so I decided to post ...
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If the primes were different, how many numbers would there be?

Given a set $S=\{s_1,s_2,\ldots\}$ of pairwise coprime positive integers greater than 1, define $T$ as the set of products of zero or more elements of $S$ so $T$ contains $1, s_1, s_2, s_1^2, s_1s_2,$ ...
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A function can provide the complete set of Euler primes via a Mill's-like constant. Is it useful or just a curiosity?

The following $f(m,n)$ function provides the complete set of Euler primes (OEIS A196230): $$f(m,n)=m^2-m+[\lfloor E^{2^n} \rfloor - {\lfloor E^{2^{n-1}} \rfloor}^2 +\frac{\lvert n-(\frac{1}{2}) \...
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Prime Powers and Differences of Consecutive Cubes

I am wondering if it has been proven that there does not exist a prime $p$ and an integer $r \ge 3$ such that $p^r = (n + 1)^3 - n^3$ for some integer $n$. Note that this is a special case of Beal's ...
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The GCD of a Univariate Integer-Valued Polynomial over a Set

Let $\mathcal{I}[X]$ denote the subring of $\mathbb{Q}[X]$ consisting of all integer-valued polynomials (i.e., $f(X)\in \mathbb{Q}[X]$ such that $f(k)\in\mathbb{Z}$ for all $k\in\mathbb{Z}$). For $f(...
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Multiplicative group of $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ is cyclic

This question has been explored thoroughly, and in more generality too. For general fields, I am aware of standard proofs. However, I was naively trying to prove it in the simple case of prime $p$ ...
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Is there a relationship between local prime gaps and cyclical graphs?

By defining the following algorithm I was able to generate some interesting graphs using the values of the gaps between consecutive primes: Start in any prime $p_i$, this will be the initial ...
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On the change $u=x^{1+\frac{1}{p_n}}$ in $\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx$, where $p_n$ is the nth prime number

In [1] Wikipedia say that for $\Re s>1$ the Riemann zeta function satisfies $$\log \zeta(s)=s\int_0^\infty\frac{\pi(x)}{x(x^s-1)}dx,$$ where $\pi(x)$ is the prime counting function, and say too ...
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Why is $p_n \sim n\ln(n)$?

I know that the prime number theorem states that the number of primes less than $x$ is approximately $\frac{x}{\ln(x)}$. However, why does this mean that $p_n \sim n\ln(n)$? (where $p_n$ is the $n$-th ...
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Primes of the form .$..55555444433322122333444455555…$

What is the smallest prime number of the form $...55555444433322122333444455555...$, where the concatenation runs through the first natural numbers, and where each decimal number $n$ being repeated/...
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Is there an advantage in using continued fractions for Catalan or Fibonacci-Lucas primality tests?

I am studying the basic theory about continued fractions and also reviewed here at MSE former questions to learn more. While reviewing the questions and answers, I found references to the Fibonacci ...
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Number of squarefree numbers and the Basel problem

Who discovered/proved that there are about $$ \frac{x}{\zeta(2)} $$ squarefree numbers up to $x$, or (roughly) when was this first known? Today I think this is considered 'obvious', but I don't know ...
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Does the set of $m \in Max(ord_n(k))$ for every $n$ without primitive roots contain a pair of primes $p_1+p_2=n$?

I have made the following observation: for those n even numbers that do not have primitive roots modulo n ,$Pr(n)$, the set $M(n)$ of those $k$ having a maximum multiplicative order $ord_n(k)$ ...
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lattice walks with primes and composites

In the regular square lattice create a walk moving according the value of a counter. Consider two types of walks: In the first walk advance forward one unit if the counter is a composite number and ...
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Regularities in a prime-exponent graph

Let $\Omega(n)$ be the number of prime factors of $n$ with multiplicity, i.e., if $n=p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k}$, $\Omega(n) = e_1 + e_2 + \cdots + e_k$ (OEIS). For example, for $n=9000 = 2^...
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Renyi entropy of prime gaps

Denote with $p_n$ the $n$-th prime number and let $$ h_N(d) = |\{ n : p_{n+1} < N, p_{n+1} - p_n = d \}| $$ be the number of times that prime gap $d$ happens for primes less than $N$. Let $H = \...
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When is a number like “ddd…ddd”+1 (where d is a digit) a perfect square or a prime?

Inspired by Is the number $333{,}333{,}333{,}333{,}333{,}333{,}333{,}333{,}334$ a perfect square?, I wonder when numbers like these are perfect squares. Certainly, all numbers of the form $000...0001$ ...
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Understanding Newman's proof of the prime number theorem

I am trying to understand D.J. Newman's proof of the prime number theorem, as presented by D. Zagier. I am not too familiar with analysis, and so there are some things I don't understand. In (III), ...