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# 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.

1,235 questions
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### How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?

Consider a system of $n$ divisibility conditions on $n$ prime variables: $$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$ where $a_{i,j}$ are bounded integers. How many solutions ...
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### A006517: Integers with $n\mid 2^n+2$

The following question was asked at Math StackExchange but, having attracted some attention, didn't get solved. Problem 323 from the Mathematical Excalibur Vol. 14, No. 2, May-Sep. 09, linked here (...
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### Wieferich primes and arithmetic prgressions

Let $p$ be an odd prime number. Let $K$ be a number field with Galois group $G$ and $H$ be a subgroup of $G$ stable under conjugation. Then the Cebotarev density theorem gives that \mathcal{L}=\{\...
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### Additive and multiplicative convolution deeply related in modular forms

From the fact spaces of modular forms are finite dimensional, from the decomposition in Hecke eigenforms, and from the duplication formula for $\Gamma(s)$ there are a lot of identities mixing additive ...
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### Why does each integer between two consecutive primes have at least one “unique” non-trivial divisor?

Why does each integer $x$ between two consecutive primes have at least one non-trivial divisor that unique on set of all integers between these two consequtive primes except $x$? We call a divisor $d$...
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### Iterated Twin Prime conjecture [closed]

"Conjecture. The sum of a twin prime pair greater than or equal to 24 can be expressed as the sum of two twin prime pairs." Examples: ...
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### Prime character sums

Let $p$ be a (large) prime number, and let $\chi : (\mathbf{Z}/p\mathbf{Z})^{\times} \rightarrow \mathbf{C}^{\times}$ be a Dirichlet character of conductor $p$. We have good estimates on the character ...
4answers
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### Can anything deep be said uniformly about conjectures like Goldbach's?

This is a soft question sparked by my curiosity about the intrinsic depth of Goldbach-like conjectures as perceived by current experts in number theory. The incompleteness theorem implies that, if our ...
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### Least (and largest) possible number of non-relatively prime pairs among consecutive integers

Given a set of positive integers consider the graph whose vertices are those integers, two of which are joined by an edge if and only if they have a common divisor greater than 1 (i.e, they are not ...
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### Consecutive integers each of which has a large prime factor

There are many results about consecutive integers all having small prime factors. But what about consecutive integers each of which has a large prime factor? More precisely, let $P(n)$ be the ...
1answer
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### Cipolla's Prime numbers function: Computing the coefficients of the polynomial

In many publications dealing with asymptotic determinations of the n-th prime number, the following paper is cited: : M. Cipolla, La determinazione asintotica dell’n-esimo numero primo. rendiconti ...
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### Why do polynomials $x^n + 1 \bmod N$ close a shorter cycle when $n$ is even than when $n$ is odd?

Polynomials $f(x) \bmod N$, where $f(x)$ is of integer coefficients and $N$ is a composite of two distinct primes $p, q$, form a cycle --- usually leaving a tail because the cycle tends to close not ...

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