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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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Why don't we just apply Fermat's little theorem directly in Miller-Rabin primality test?

Fermat's little theorem states that for $2$ integers $a$ and $n$ such that $n$ is a prime that doesn't divide $a$, $a^{n-1}\equiv 1\pmod{n}$. So in Miller-Rabin test, if we want to test the primality of an integer $p$ (assuming $p>2$, $p$ is odd,…
Mohamed EL Tair
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For every positive even number $n$, does there always exist a pair of prime $(p,q)$ such that $p-q =n$?

$$2 = 5-3 \\ 4 = 7-3 \\ 6 = 11-5 \\ 8 = 19-11 \\ 10 = 13-3 \\ \vdots$$ For every positive even number $n$, does there always exist a pair of prime $(p,q)$ such that $p-q =n$?
with-forest
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de Polignac conjecture and Wilson theorem

de Polignac conjecture: For any positive even number $d≥2$, there are infinitely many prime gaps of size $d$. We know (https://mathoverflow.net/questions/339781/generalization-of-wilsons-theorem-for-prime-tuples) that $(k,k+2)$ forms a twin prime…
Safwane
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Prime properties

Are there any properties that all primes share in common (while all the non-primes don't share it), and could be written in a simple formula which involves only 1 positive integer variable? Thank you!
user1398593
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Can this statement on pernicious numbers be true?

From Wikipedia "In number theory, a pernicious number is a positive integer that has a number of 1s in its binary representation which is a prime number. Equivalently, the sum of the digits of a pernicious number, when represented in base 2, is a…
CBusBus
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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 pairs of primes seem to get arbitrarily close…
Justin Tang
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What should a subset of prime numbers $S$ fullfill for the product of its elements plus one to equal another prime number?

If we let $S$ be any subset of the set of prime numbers $P$, for some S, the product of the elements of $S$ plus 1 is, in fact, another prime number: $$\exists S \subset P:[\prod_{i=1}^{|S|} S_i] + 1 \in P$$ Here are some examples to the…
usuyus22
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How many numbers below a product of first n primes P are coprime with P?

Products of first primes are P1= 2; P2= 2*3=6; P3= 2*3*5=30; P4=2*3*5*7=210; etc. Coprimes. P1 : {1} P2 : {1,5} P3 : {1,7,11,13,17,19,23,29} Coprime count. P1: 1 P2: 2 P3: 8 Pn: Cn What is the sequence of coprime count Cn?
Stepan
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Removing every Nth number with many different N, does this equation already exist?

Removing every 2nd integer of all numbers, removes 50% of all numbers Removing every 3rd integer of all numbers, removes ~33% of all numbers Removing every 2nd and 3rd integer of all numbers, removes ~66% of all numbers The equation I've come up…
Harry Svensson
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3 primes conjecture

let be $ p,q,r $ prime numbers AND 'n' an integer is then true that we can always look for p,q,r and an integer n so $$ p^{n}+q=r $$ $ 5+2=7$ $ 2^{3}+3=11 $ $ 3^{4}+2=83 $ abnd so on
Jose Garcia
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can $K, K+l, K+2l,...,K+l^2$ all be primes if $K$ is prime and $l$ $is$ $even$?

Given a $K$ prime number and $l$ $is$ $even$ that satisfy $l
Shaq
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Probability of Prime within radius around number

Here is my question: do we have any kind of estimate about $p_{k, d}(n)$ the probability that there are at least $k$ prime numbers in a radius of $d$ around $n$? Do you have any suggestions regarding related work? For instance, we know that for $n$…
ted
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Problem over prime numbers

Which is the largest integer $n<1000$ so that $n$, $n+2$ and $n+4$ are primes? I have tried to solve this problem but have not reached an argument worth
Roiner Segura Cubero
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Can $x^n-(x-1)^n$ be Prime if $n$ is Not Prime?

I'm hoping someone can provide an answer or a link to a proof regarding this question. Edit: The question has been put on hold because I did not expound on why the answer to this was of interest to me or the community, so, even though I've received…
Jaarbahd
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Math expression for an infinite sequence of primes

At the beginning I would like to ask if there are infinite prime numbers of the form: $$\prod_{i=1}^{n} p_i + 1$$ where $p_i$ is the $i$-th prime number; but after a google search I found that they are called Primorial primes and it is not known if…
Vor
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