# 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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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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### 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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### 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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### 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 $1, s_1, s_2, s_1^2, s_1s_2,$ ...