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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.

A prime number (or a prime) is an element of the greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number ... The fundamental theorem of arithmetic establishes the central role of primes in :

Any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.

Here you get the first 50 millions of primes.

The concept of prime numbers is extended in ring theory, where an element $p$ of a ring $R$ is prime if and only if whenever $p\mid ab$, then $p\mid a$ or $p\mid b$.

One can easily see that this extends the definition of prime numbers in the natural numbers.

12562 questions
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Property for 4 consecutive positive integers?

I derive the answer G by randomly choosing four integers multiple times, which is a bit time-consuming for ACT Math. Could anyone offer me a quicker method to cope with it?
HypnoticBuggyWraithVirileBevy
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Need help understanding why formula for prime number is correct

I was reading a Mathematical paper from the '40s, and came across the following formula: $$q = \theta\sqrt{\frac{n}{\nu}}$$ Where $q$ is a prime number, $\nu$ is a given integer, and $\theta \in (1, 1+\vartheta)$, where $\vartheta$ is a given real.…
Xander L
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Prime numbers of certain form

Are there infinitely many prime numbers of the form $p^{2h}+p^h+1$, where $p$ is a prime and $h$ is a positive integer?
Mathsa
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$p * z - q * n = 1 \implies \gcd{(z,n)} = 1$

Prove that: let n $\in \mathbb{N} $ and $ z \leq n \in \mathbb{Z}$ $\exists p,q \in \mathbb{Z}: p * z - q * n = 1 \implies \gcd{(z,n)} = 1$
Another Noone
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Would consecutive prime integers be 2 number's apart like 1,2,3 and 5?

I know it's kinda stupid but I would like to learn about integers, asking here cause I couldn't find it online. Just want an explanation, please.
Isaiah Stuart
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Is $-a^2+2ta-1$ and $-b^2+2tb-1$ are also coprime for any positive integer $t>a$ and $t>b$

Let $a,b$ be two coprime positive integers. I am asking if $-a^2+2ta-1$ and $-b^2+2tb-1$ are also coprime for any positive integer $t>a$ and $t>b$.
Safwane
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Maximum possible number of primers in an interval

i need to know the maximum possible number of prime numbers between $10n$ and $10n+10$ for $n>0$. So far I've found the only possible primes are $10n+1$, $10n+3$, $10n+7$, and $10n+9$ but there are values of n where some of these are not prime. I've…
CoolJedi132
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Can I say "Any number, including prime themselves, is composed of the addition of primes"?

Since prime numbers are like irreducable blocks in the number system, can I say that: "Any number, including prime themselves, is composed of the addition of primes?" I have quickly checked for a few numbers up to 10 and could tell that it is True.…
Yi Xiang Chong
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Is $\frac{2(4^n-1)}{3(2n+1)}$ always an integer if $2n+1$ is prime?

$$\frac{2(4^n-1)}{3(2n+1)}$$ it's all in the title.. I have just seen this on a forum, and I wondered whether it is true, and why. I'm not well versed in number theory. I tried to find a counterexample using c++, but the number becomes very large…
GDGDJKJ
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Primes equivalent to 1 (mod 8) Proof

Show there are infinitely many primes that are equivalent to $1 \pmod{8}.$ I've tried using proof by contradiction: if there are only $n$ primes equivalent to $1 \pmod{8}$ with a product of $P$, then $(2P)^4 + 1 \equiv 1\pmod8.$ How do I finish…
Robert Mugabe
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Can the mean of 2 consecutive prime numbers be prime?

This is apparently a "hard" question, and I don't know if I'm missing something, but it seems trivial to me. Aside from 2, all other prime numbers are odd. So the mean of any consecutive prime number is of course even and hence divisible by 2. So…
David
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Inequality For Asymptotic Prime number relation

Following from the asymptotic relations described here, I am seeking a proof or minimal counter example for the inequality below, the right hand side of which occurs for $n=9$ and $k=2$. $$n-\Biggl(n^k -{\lfloor n^{\frac{1}{k}}…
Adam Ledger
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Does the proportional error $n\ln(n)$ for the $n$th prime approach a non-zero limit?

The following graph shows the proportional error of $n\ln(n)$ for the $n^{th}$ prime. The proportional error appears to fall towards $0.1$, and not zero, when considering the first $n=10,000,000$ natural numbers. Is this a correct conclusion, or…
Penelope
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Primes separated by multiples of 4

Q. Is it the case that for every prime $p$, there is a larger prime $q$ such that $q = p + 4 n$, $n \ge 2$ ? For example: $5 + 8 = 13$, $13 + 16 = 29$, $29 + 8 = 37$, and so on. I came upon this constructing a stacked version of Ulam's…
Joseph O'Rourke
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Proof that $a^p-a$ is divisible by $p$ where $p$ is a prime number?

What is a proof that $a^p-a$ is divisible by $p$ where $p$ is a prime number?
ASHOK KUMAR SAHU
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