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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As $n$ grows sufficiently larger, $\pi(n)<\pi_{1}(n)$, where $\pi(n)$ and $\pi_{1}(n)$ is the number of prime and semiprime $\leq{n}$, respectively

From $P_{12}=37$ the number of semiprime(s) appears to be higher than the number of prime(s). Though I couldn't check for a higher $n\geq{500}$ for several limitations, I could really use any proof or ...
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A conjecture about the prime function $p_n$

While testing my system Zet for computational mathematics I find possible relations now and then. The latest is: Conjecture: For all $(m,n)\in\mathbb Z_+^2$ except $(3,4),(4,3) \text{ and } ...
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Prove that if $p|a-b$ then $p^{n+1}|a^{p^n}-b^{p^n}$

I need help with the following problem, I don't know how to continue. Let $p$ be a prime. Prove that if $p|a-b$ then: $$p^{n+1}|a^{p^n}-b^{p^n}$$ At first I thougt the following: $$p|a-b$$ $$a ...
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Every even integer $n>2$ is a semiprime or sum of two semiprime numbers.

Progress: A slightly stronger version of the original assumption is this: Every even integer $n>2$ is a semiprime or sum of two even semiprime numbers. I was wondering as to how this ...
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Interesting and unusual word problem with prime numbers and factors

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with prime numbers, but other than that, the textbook gave no hints really and ...
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Galois Group of Splitting Field, $S_4$

I've shown that the polynomial $x^4+px+p \in \mathbb{Q}[x]$, where $p$ is prime, is irreducible by Eisenstein's criterion. However, it remains to be shown that the Galois group of the splitting field ...
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A Product of Primes [on hold]

Anyone ever seen a function like this: F(1)=1st prime, F(2)=2nd prime * 1st prime, F(n)=nth prime * F(n-1)?
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On a certain prime structure.

It is unknown whether there are infinite primes $p$ where $2p-1$ is also a prime. Is it known there are only finitely many primes $p$ such that both $q$ and $2p-1$ are primes where $p-1=2aq$ for any ...
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Range to look for first $N$ prime numbers.

What range of numbers $[2, X]$ should I search, to be absolutely sure that I would get exactly or more than $N$ prime numbers within that range? Any formula for $X$?
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When is a prime $p$ a quadratic residue modulo $3$?

Simple. When $p \equiv 1 \pmod 3$, it is a quadratic residue, and when $p \equiv -1 \pmod 3$ it is not a residue. So can we have a nice expression for the Legendre symbol $\left(\frac{p}{3}\right)$? ...
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Proof — Infinitely many primes of the form $4k + 3$ — origin of $4(p_1…p_k - 1) + 3$

I know there are sundry questions — like this pdf — and this (10.) Prove that any positive integer of the form $4k + 3$ must have a prime factor of the same form. Because $4k + 3 = 2(2k + 1) + 1$, ...
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A conjecture about primes

Let $p_n$ be the $nth$ prime and define $p_n^{(m)}$ by $p_n^{(1)}=p_n$ and $p_n^{(m+1)}=p_{p_n^{(m)}}$: $p_n^{(2)}=p_{p_n}$, $\;p_n^{(3)}=p_{p_{p_n}}$ and so far... For some coprime numbers $a,b$, ...
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How to prove that $p$ divides $a^p -a$ for every integer $a$. [on hold]

How to prove this Fermat's little theorem: $p$ divides $a^p -a$ for every integer $a$.
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Is it true that a cycle with a period of 29 hours over 24 hours leads to a non-recurring pattern and how to prove it?

The default 'reset time' for Internet Information Services is 29 hours. The reason for this is that 'Wade [person on the team who developed the setting] suggested 29 hours for the simple reason ...
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Arithmetic sequence whose any five consecutive elements contain a prime

Consider an arithmetic sequence $\{11 + 13k : k\in\mathbb{N}\cup\{0\} \}$ Does this sequence contain five consecutive composites? If we look at some selections of five consec. elements: $$11, 24, 37, ...
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How to solve difficult positive integers and co-prime word problem?

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with derivative of algebra and prime numbers, which yields the shortest, ...
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Proof Enquiry, Field of order $p^n$ [duplicate]

I want to prove that there exists an inclusion $\mathbb{F}_{p^a} \hookrightarrow \mathbb{F}_{p^b}$ iff $a \vert b$. Suppose that $a \vert b$, then $b =ac$ for some $c \in \mathbb{Z}$. Consider then ...
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Is there a name for these primes?

What is the name for primes $p$ where $2p-1$ is also a prime? $2p+1$ is a Sophie Germain prime. On average if $p$ is a primes how many primes of form $2p^n-1$ could we expect where $0<n<B$ ...
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Proving primality of $p$ without making any calculation involving $p$ directly

Wilson's Theorem states that a positive integer $p > 1$ is prime if and only if $(p-1)! \equiv -1 \pmod p$, showing a relationship between factorials and prime numbers. Finding it curious, today I ...
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what is Prime Gaps relationship with number 6?

Out of the 78499 prime number under 1 million. There are 32821 prime gaps (difference between two consecutive prime numbers) of a multiple 6. A bar chart of differences and frequency of occurrence ...
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Factorial and primorial twin primes

Factorial primes are are primes of the form $n! \pm 1$ and primorial primes are primes of the form $p\#\pm 1$, where $p\#$ is the product of all primes $\leq p$. To cite ...
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Number of primes of a certain form

Let $p_n$ be the nth prime. Are there an infinite number of primes of the form $2p_n+1$? Is something known about questions like this?
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Is that true that all the prime numbers are of the form $6m \pm 1$?

Is that true that all the prime numbers are of the form $6m \pm 1$ ? If so, can you please provide an example? Thanks in advance.
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Possibly New Prime Conjecture

I was in the midst of proving a conjecture when I came across an observation that led me to forming a potentially new conjecture. The conjecture goes as follows: Any given sum of twin primes ...
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How do I make this formula for the primes more concise?

The form I made for the $(n+1)^{th}$ prime $p_{n+1}$ is $\displaystyle1+\sum_{j=1}^{2p_n-1}\lfloor\frac{p_n!^j}{j!}\rfloor-\lfloor\frac{p_n!^j-1}{j!}\rfloor=p_{n+1}.$ Problem is, just like any ...
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Do primes “behave” in this way?

Suppose that we choose some real number $\varepsilon >0$. Can we always find $n_0(\varepsilon) \in \mathbb N$ such that for every $n> n_0(\varepsilon)$ there is a prime number $p$ such that ...
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1answer
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Show that any arithmetic progression contains a sequence of composites of arbitrary length

My question is inspired by this one: Arithmetic sequence whose any five consecutive elements contain a prime A more precise form: Let $(x_n)|_{n=1}^{\infty}$ be an arithmetic progression such that ...
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2answers
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Fractals using just modulo operation

Let us calculate the remainder after division of $27$ by $10$. $27 \equiv 7 \pmod{10}$ We have $7$. So let's calculate the remainder after divison of $27$ by $7$. $ 27 \equiv 6 \pmod{7}$ Ok, so ...
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Highest ratio between consecutive prime numbers

Let $r = p_2/p_1$; where $p_1$, $p_2$ are consecutive prime numbers. What is the highest possible value of $r?$ Are there any consecutive prime numbers such that $r > 5/3$?
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Solutions to set of equations involving prime numbers

Is there a collection of distinct positive integers $(k_1, k_2, k_3, p_1, p_2, p_3)$ such that: $p_1, p_2, p_3$ are odd primes, and $k_1, k_2, k_3$ are odd $(k_1 + 2) p_1 = k_2 p_2$ and $(k_2 + 2) ...
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$\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$

Conjecture For $n \ge 1 $ , $m \ge 1$ $\pi\left(\left(n+m\right)^2\right) - \pi\left(n^2\right) \ge 2 \cdot m$ where $\pi\left(n\right)$ is the prime counting function . Does this conjecture ...
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How far to nearest/next prime?

Is there is metric to know how far we are from the nearest prime number. For example if my number is 38, then we are 3 numbers away from 41? If such a metric doesn't exist, is there an upper bound ...
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Express a prime $p$ as $p=a^2-2b^2$

Suppose $2$ is a quadratic residue modulo $p$ for an odd prime $p$. That is, there is an element $u$ such that $u^2 \equiv 2 \pmod{p}$. From here, can we prove that there exist integers $a$ and $b$ ...
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Do all primes occur in some sequence associated with the Collatz conjecture?

Let $f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases}$ For an arbitrary prime $p$ are there some start value $x_0$ such that $p = x_k$ for some ...
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Roots of the equation $x^2+1=0$ in $\Bbb Z/p^{n}\Bbb Z$

Let $p$ be an odd prime number and $n$ be a positive integer. I want to consider roots of the equation $x^{2}+1=0$ in the ring $\Bbb Z/p^{n}\Bbb Z$. Suppose $n=1$. Find a condition on $p$ such ...
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Rabin Miller test [closed]

I am working on rabin miller test. What amaze me, is its showing some prime number composite like this one 3494128965953(which is prime but test fails) with 100 iterations.With testing of composite ...
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Prime numbers and arithmetic progressions

Whether there exist a polynomial $f$ such that for every $n$ there exist prime numbers $p_1, \ldots, p_n$, and an integer $b$ such that every $p_i$ and $b$ are less than $f(n)$ and $p_1×\ldots×p_n×b + ...
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Random binary array shows patterns around prime numbers

First post, so please let me know if I'm doing something wrong or if this question does not belong here. I have been toying with java to visualize an interesting 2D binary array I thought of today in ...
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Twin prime conjecture proof error

I am absolutely sure this is wrong but I can't find why. For every integer $n$ there exist a finite number of primes less than $n$. Take the set containing those primes and multiply them together to ...
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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 ...
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Proof of the following property about prime numbers

Is there a proper proof of the following property: Let $p$ be a prime number. The number of invertible elements in $\mathbb{Z}/p^n\mathbb{Z}$ is $(p-1)p^{n-1}$.
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$\mid-1+2-3+4-5+6-7+10-11+…+(p_{k}-1)-p_{k}\mid=k$? Where $p_{k}$ is the $k$-th prime.

I'm not sure if it's a Telescoping series but I tried the generating rule to prove and test the series but I'm not getting any insight and I got stuck. Here are few Examples: ...
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Showing the infinitude of primes using the natural logarithm

I came across this proof in Proofs From the Book by Aigner and Ziegler. It uses the inequality $logx \leq \pi(x)+1$. (Here, we use natural logarithm) The proof starts with the inequality $log$ $x ...
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Given a number find other numbers diagonal to it in ulam spirals

Given a number, is it possible to find the 4 numbers diagonal to it in an Ulam Spiral? For example: given the number 22, is there some formula that will allow me to find the numbers that lie ...
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A number theory problem?

Let $p$ be a prime number. $k>1,n$ be integers and $0<λ<p^k$ be integer. Is there any $λ,n$ s.t the following relation satisfies: $$(2^n−1)(p^k−λ^2)=p^k(p^k−1).$$ I guess there isn't but I ...
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If $p > 3$ is prime, then $12 $ divides $p^2 - 1$

First up, I know there are a lot of similar questions with 24, not 12. So bare with me please :) What is the Question? Consider the following numbers of the form $p^2 - 1$ where $p$ is prime. $$5^2 ...
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Polynomial problem involving divisibility, prime numbers, monotony

Let $f$ be a polynomial function, with integer coefficients, strictly increasing on $\Bbb N$ such that $f(0)=1$. Show that it doesn't exist any arithmetic progression of natural numbers with ratio ...
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Uniquely identify any finite subset of an infinite set

Let $U$ be an unbounded subset of $\mathbb{N}$. Let $D = \mathcal{P}_{<\omega}(U)$ (the set of all finite subsets of $U$). Let $f$ be an injection such that: $f: D \rightarrow \mathbb{N} $ ...
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Sum of products of $(1 - 1/p)$

Let $\pi(n)$ denote the number of primes not greater than $n$, and $p_k$ the $k$th prime, so that $p_{\pi(n)}$ denotes the largest prime not greater than $n$. I'm interested in the value of the ...