# 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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### What does it mean to count a group of numbers with their multiplicity?

In this question someone previously asked They presented the problem: Given that the number 8881 is not a prime number, prove by contradiction that it has a prime factor that is at most 89. One ...
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### Primes of the form $a^2+b^2+c^2$, $0<a<b<c$ [closed]

Are there results showing which prime numbers that can be expressed as the sum of three different integers greater than zero? By the three square theorem of Legendre a natural number can be written ...
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### Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let $M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}}$ ...
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### Division of prime numbers

a) Let p and q be different odd primes. Is there any such numbers for which $$(p-1)(q-1) \mid (pq)^2 +3$$? b) Let q be an odd prime number. Is there any q for which $$6(q-1) \mid 81q^2 + 3$$? (I ...
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### Is the largest gap between consecutive primes less than the first $27,000$ integers equal to $52?$

Is the largest gap between consecutive primes below the first $27,000$ integers equal to $52?$ At what point does a gap greater than $52$ occur? I tried analyzing a formula due to Maynard, Tao, and ...
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### Prove these two statements are logically equivalent

Consider the following two statements about non-negative integers: Goldbach's conjecture: Every integer > 1 is the average of two primes SumOfThreePrimes: Every integer > 5 is the sum of three primes....
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### A Simple Proof of the FTA using only elementary theory?

UPDATE/WARNING: DO NOT READ (WASTE OF TIME) The effort I put in here is now an embarrassment as it goes nowhere. I can't delete this posting since there is an answer, but if a moderator could delete ...
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### The pattern of $n$-th prime minus its reversal in even and odd bases

I recently saw the sequence A265326 on OEIS and also in Brady's Numberphile video Amazing Graphs ft. Neil Sloane The sequence is such: Start from $2$, given the $x$-th prime number and convert it ...
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### Best possible estimate for $n$th prime [duplicate]

We all know the $n$th prime can be approximated by. $nln(n)$ by Prime number theorem. We also got inequalities for the estimate . But my question is what is the best possible estimate for $n$th ...
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### $n^{th}$ Dimensional Prime Nexus Conjecture (Hamilton's Path & Cycle)

The system and proposition of Prime Nexus - We have to arrange $n>1$ Distinct natural numbers in a sequence such that the adjacent elements sums upto any Prime number. Convenience is when we set ...
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### Algorithm to decompose a number into the product of an integer and a base two exponencial

So i`ve been asked to code an algorithm that decomposes an integer into the product of a base two exponencial and some integer.Something like number = k.(2^n) , k and n being random integers with k ...
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### What is known about the counting function of Gaussian primes"

The counting function of primes among $\Bbb{N}$, describing the asymptotic density of the primes, is well known (the Prime Number theorem). Let's define a mild generalization of the counting function ...
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### time complexity of some specific system of $(n-3)/2$ linear equations with $(n-3)/2$ unknowns.

let take n a odd integer greater than 3 if I have to solve this system of $(n-3)/2$ linear equations with $(n-3)/2$ unknowns to find some propriety about the number n, here is formula for every ...
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### Generalization of digit-wise divisibility lemmas [duplicate]

Background In undergraduate Abstract Algebra homework, for an integer $n$ with decimal representation $a_m a_{m-1} ... a_1 a_0$, I proved that $3$ divides $n \iff 3$ divides $\sum_{i = 0}^{m} a_i$, ...
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### Confusing thing at the proof of proth theorem

$Proth$ $theorem$ simply depends on a result which proved by pocklington ; The result says : Let $N-1=q^nR$ where $q$ is prime, $n\ge1$ , and $q$ doesn't divide $R$. Assume that there exists an ...
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### Are Wieferich primes Wieferich numbers?

In the article 'On a conjecture of Crandall concerning the $qx+1$ problem' by Franco and Pomerance, they define Wieferich primes to be prime numbers $p$ for which $p^2|2^{p-1}-1$ and Wieferich numbers ...
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### Is there a set of primes such that the union of their multiples occurs with frequency $1/n$, where $n$ is an integer?

Are there two or more primes, $p_1, p_2, \ldots$ such that the union of their multiples occurs with frequency $1/n$, where $n$ is an integer? If so, can that n be prime? I put it into a computer and ...
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### An interesting algorithm about prime numbers that I thought today

I thought up the following algorithm today: Choose $a_1\in\mathbb{Z}^+\setminus\{1\}$. Then let $a_{n+1}=a_n+p_n$, where $p_n$ is the largest prime factor of $a_n$. The algorithm is easy, but I ...
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### Why such an interest for the error term in the Prime Number Theorem

I have some issues when dealing with people working outside number theory, to motivate and justify in some sense the problems I am interested in. Mainly, here are the issues I do not know enough ...
I know that under GRH we have $$\sum_{\substack{1 \leq n \leq X \\ n \equiv a (q) }} \Lambda(n) = \frac{X}{\phi(q)} + O(X^{1/2} (\log X^2)).$$ From this I would like to deuce a abound for $E$ ...
Can someone help me to figure out what $$\lim_{n\to\infty} \frac {c_n}n-\frac{c_n}{p_n}-\frac {c_n}{n^2}$$ is equal to? I am pretty sure its $1$ and i tried many different things but i couldnt figure ...