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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.

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Is there an upper bound on a prime between $2n$ and $3n$?

It is known that there exists a prime $p$ between $2n$ and $3n$. I'd like to know whether there is an upper bound on $p$ or whether there is an upper bound on a prime between $2n$ and $kn$, where $k$ is an odd integer.
Jiantao Li
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Can anyone please determine integral below?

I was creating a paper on P.N.T but I stucked here so,
Shivanshu
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Is there a positive integer number which is not prime and which has exactly two positive integer divisors?

For starters, I am not a mathematician, but I was fascinated by prime numbers for many years. So today, I came up with the definition which I am extremely curious about, because I believe it poses a mathematical problem. Prime number is a positive…
Arislan Makhmudov
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Given $2$ integers $a,b$ which sum to a prime, is it possible to find a new integer $x$, so that $a+ bx$ is NOT prime?

We start with $2$ unknown integers $a,b$. It is known that the sum of $a$ and $b$ is prime and $a$ is not negativ and $b$ is greater than $0$. $a+b$ is prime, where $a \ge 0, b > 0$ Is it possible to find a non negativ integer $x$ depending on $a$…
Lantanar
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Proof that $41$ does not divide $a^5 - 2$

How can i proof that $41$ does not divide $a^5 - 2$ for any $a \in \Bbb Z$? I think i must show that $a^5 - 2$ is multiple of 41. So if i do $(a^5 - 2) / 41$ i must get 0 rest? I'm stuck sorry. Thank you.
Mudasty
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Need help to prove that any non prime number that >= 4 can be written in the form: i^2 + i * k

Any non prime number that >= 4 can be written in the format: i^2 + i * k. Condition: i>=2, k >= 0 (i, k are integers) 4 = 2^2 + 2*0 -> Good 6 = 2^2 + 2*1 -> Good 10 = 2^2 + 2*3 -> Good ... Thanks!
LHA
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About the existence of a form but with a finite number of primes

I see this page (https://en.wikipedia.org/wiki/Category:Classes_of_prime_numbers) about different forms of prime numbers. I see also, that it was conjectured that all these forms contain infinitely many primes. My question is about the existence of…
Safwane
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Number of primes that can divide ...

Find the number of primes $p$ less than $100$ such that $p$ divides $x^2+x+1$ for some positive integer $x$. I do not understand how to approach this problem. Is there a formula I need to use? I don't need the answer.
Electiwirez
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What is the Largest Possible Prime Number? sic The largest possible prime set?

Okay, "What is the Largest Possible Prime Number?" First it appears this question is outside the bounds of good mathematical reasoning for two main reasons. 1) The concept of 'largest possible prime' is ill-defined. 2) The concept also appears to…
Moga
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I have theory in prime number and I want to help me for proof it

Prove the following: If we take $n$ when $N$ is natural number bigger than one, then: For any value for $n$, there are $k$ number which causes the following equation to be true $$n^2-k^2=p\times q$$ where $k$ is a natural number, $p$ and $q$ are…
small
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An equivalence.

Where p is a prime number Prove that: $$(\forall \left( a;b\right) \in \mathbb{Q} ^{2}) ;a+b\sqrt {p}=0\Leftrightarrow a=b=0$$ Notice that :$$\sqrt {p}\not\in \mathbb{Q}$$
Am ine
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Prove That $x^2-1$ is never prime, given X is an odd number.

How do we prove that this expression is always 'not prime'?
user405925
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Series of consecutive composite numbers

Being given $a \in \mathbb{N}$, find the smallest $N$ such that $\{ N, N+1, N+2, N+3, N+4, ...., N+a\}$ are all(consecutive) composite numbers. For example for $a=2$, $N=8$ ; for $a=4$, $N=24$. I am looking for a general formula.
Amir
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$p^2+p+1$ is either prime or semi prime, if $p$ is prime number

I am trying to come up with conjectures related to prime numbers and came up with this question. The last question I asked was wrong. So for this question I have tried numbers till $30$. Is this conjecture valid?
jnyan
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Is there any formula which gives better approximation than this formula?

Let $g(n),f(n)$ be functions of $n \in \mathbb{N}$ such that $g(n)=(n−1)^\frac{1}{n−1}$ and $f(n)=\frac{a(g(n)^n)+(g(n)+(\frac{b}{n}))^n}{2}$. Define $P(n)=(f(n))\log_e (f(n))$. $P(n)$ gives the $n$th Prime Number. $a=0.520372$ , $b=0.544973$ This…
Ashish Goel
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