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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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What do we know about the distribution of Mersenne primes?

Mersenne primes are primes of the form $M_n = 2^n - 1$. I'm wondering how far apart successive Mersenne primes can be. For example, is $M_{n+1} \le O((M_n)^e)$? Or, is $M_{n+1}$ always less than some power of $M_n$? If not, how close together…
Matt Groff
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smallest prime number made of consecutive numbers as its digits in decimal representation

i have been curious about prime numbers lately and have been wondering what is the smallest prime number that is made of consecutive numbers as its digits. example of number for further clarification: 1234567891011121314151617......... i have…
bscharan
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Explicit formulas for primitive roots?

For a Fermat prime or an "upper" Sophie Germain prime a primitive root is explicitly known. Are there further results when the factorization of p-1 is known? Is it unlikely that we ever get explicit formulas for larger classes of primes, not only…
Landei
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Calculating prime numbers

I have a program which determines if a number is prime or not. The basic algorithm simply checks for the number being divisible by 1 and itself. My question is, is there an upper limit to checking numbers for divisibility without a remainder? If…
P.Brian.Mackey
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What are primes in the form of $2^n+1$ called?

What are primes in the form of $2^n+1$ called? I know that those of form $2^n-1$ are Mersenne primes, but I'm not sure about the other ones.
ThePiachu
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For which $p\in\mathbb{P}$ does $x^4+4=p$ have solution in $\mathbb{N}$?

Find all primes $p$ for which equation $$x^4+4=p$$ has solutions in set of natural numbers. I already have my own idea in mind, though the proof seems kinda weak to me at some points. But I'm more interested in different approaches to this problem…
Lazar Ljubenović
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Is there prime number of the form$1101001000100001000001.....$after the trivial one $k(0)=11$?

Let $k(0)=11$ $k(1)=1101$ $k(2)=1101001$ $k(3)=11010010001$ $k(4)=1101001000100001$ And So on.... I've checked it up to $k(120)$, and I did't find anymore prime of such form. Are there anymore prime numbers of that form ? (I just realized that…
Ken Raosz
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Prime gaps distribution

It is well-known that gaps between successive primes have i.e. multimodal distribution (with peaks at $6 k$): I'm interested to know: what is the most suitable approximation for such weird distributions? The envelope of peaks looks like $\chi^2$ …
lesobrod
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$2^{prime} = {prime}*x+1$

I've stumbled across something I think is true, but I don't know how to show it. It started to haunt me. My conjecture is that for every prime $p$, if you look at $2^{p-1}-1$ this is divisible by $p$. For example: $p=7$. $\frac{2^6 - 1}{7} =…
Casper Thalen
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For how many positive integers $n$ is $4^n − 1$ a prime number?

This was the initial question and one of the methods to work it out include this: The second method uses the fact when $n$ is a positive integer $x − 1$ is a factor of $x^n − 1$, since, for $n ≥ 2$, $$x^n − 1 = (x − 1)(x^{n−1} + x^{n−2} + \ldots + x…
Zubair Mahmood
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Largest prime number $p$ that cannot be written as the sum of three composite numbers?

This question is obvious in the sense that prime gaps on average are larger as the numbers go towards infinity. However, I don't have any idea on how to begin tackling this question apart from very basic trial and error. Any ideas will be useful.…
Haikal Yeo
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Difference Between Primes

We know that there are two prime numbers that have a difference of one: 2 and 3. And we know there is at least one pair of primes with a difference of two: 5 and 7. Same with a difference of three: 2 and 5. This pattern continues until we reach a…
Ben
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No prime number between number and square of number

Find the values of $x \in \mathbb{Z}$ such that there is no prime number between $x$ and $x^2$. Is there any such number?
Batakj
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What is the largest 'sequential' prime number?

New to this, sorry for bad formatting but: The largest know prime number is a Mersenne Prime Number, i.e. it is in the form $2^n-1$. Whilst this is big, the last known prime number before this was millions of digits shorter. Take the list $2, 3, 5,…
Cryogenic
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Is there way to know if ODD number can be expressed as sum of two primes?

I'm solving problem and I need your help, I know that every even integer can be expressed as sum of two primes and every integer can be expressed as sum of three primes. (for all integers <= 2 * 10^9) But I want to know is there a way to check can…
someone123123
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