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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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There exist three integers $r,l,w$ such that $rx^{n}+ly^{m}+wz^{q}=1$

We say that three positive integers $x,y,z$ are coprime if there exist three integers $a,b,c$ such that $$ax+by+cz=1$$ How one can prove that $x^{n},y^{m},z^{q}$ are also coprime, i.e., there exist three integers $r,l,w$ such that…
Safwane
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remainders of three consecutive primes when product of each two is divided by the third

Take three consecutive primes p,q,r and find the three remainders for $$p\cdot q\mod r$$ $$p\cdot r\mod q$$ $$q\cdot r\mod p$$ Is it possible for all three remainders to be prime?
J. M. Bergot
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Barchart of Differences of Prime Numbers. - Why do multiples of six occur more frequently?

Differences between Primes - First Million Primes Looks like 6,12,18,24 etc. are peaks compared to their near neighbors. Any insights why? Click on https://i.stack.imgur.com/QyCQi.jpg to see Bar Chart
Michael Bloomberg
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Find all $p$ is prime number such that Both $p+14$ and $p+20$ are Prime number.

Find all $p$ is prime number such that Both $p+14$ and $p+20$ are Prime number. I knomw $p=3,p=17,p=23$, but I can't to show that is that all.
Hung nguyen
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Prime number theorem approximations.

Is $\pi(n)=\left\lfloor \frac{n}{\log n} \right\rfloor$ for infinitely many $n$? If so, are there any conditions that a set or progression contains infinitely many such $n$'s? Do they have a distribution or any properties?
user784203
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Conjecture or theorem on addition of prime numbers and a constant which sum is another prime

I am looking for theorems or conjectures that may exist regarding adding prime numbers. In particular, about "adding two or more primes, plus a fixed value, that results in a prime number". I am observing an interesting pattern with a set of numbers…
edaus
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Unable to find logic behind question on primes.

Let $p$ be a prime number and $s$ be a positive integer. Show that for any $i \in \{0, 1, . . . , p^s− 1\},\ \binom{p^s−1}i \equiv (−1)^i(mod\ p)$. Formula-wise the question attempts to take the ratio of: $\frac{(p^s−1)!}{i!(p^s−1-i)!}$, for all…
jiten
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Show that not every number of the form $N = (p_1 p_2 p_3 \cdots p_n) + 1$ is prime, where $p_1, p_2, p_3,...,p_n$ is the list of all prime numbers?

How would you show that not every number of the form $N = (p_1 p_2 p_3 \cdots p_n) + 1$ is prime, where $p_1, p_2, p_3,...,p_n$ is the list of all prime numbers? I have tried several proof techniques including the proof of infinitely many primes…
DYBnor
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Digits sum of prime number, written in ternary system

Recently, I've observed that if we write a prime number (greater than 2) with base 3 (in the ternary system), then the sum of digits will be an odd number. Is this previously been observed? Is this useful somehow? Examples: (decimal prime, ternary…
komorra
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A Conjecture upon linear expressions generating prime numbers. Is it hard to solve and easy to check?

My question will directly refer to the P vs NP Conjecture. It asks whether every problem whose solution can be quickly verified can also be solved quickly. With the aid AKS primality testing we can organize primality testings in some polynomial…
Crystal Wolftron
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Gamma function in Taylor series

The formula I am having some problems is this one $$ f\left( x \right) =\sin \left(\frac { 2(x-3)!-x+1 }{ 2x } \pi \right) =\sin \left(\frac { 2\,\Gamma (x-2)-x+1 }{ 2x } \pi \right) $$ Although this might seem too general it is a variation of…
user1398593
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Add a square to $853$ to form another square

Given that $853$ is a prime number, find the square number $S$ such that $S + 853$ forms another square number. I have no idea how you would find $S$, and trial and error doesn't help. Is there a way of finding $S$?
AOD
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For $x$ and $y$ are coprime. Prove that $xy$ and $(x+y)$ are coprime.

Given That $x$ and $y$ are coprime integers. Prove that $xy$ and $(x+y)$ are coprime.
AnisLaib
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How can tell if there is a prime number between any two numbers?

Is there a conjecture, statement, formula or anything that determines whether or not there is a prime between two numbers? I'm not looking for a computer algorithm. Any help would be appreciated.
Tobi Odeyemi
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Find all prime numbers $p$ and $q$, such that $7p+q$ and $pq+11$ are also prime numbers.

Find all prime numbers $p$ and $q$, such that $7p+q$ and $pq+11$ are also prime numbers. Based on the fact that all primes, besides 2, are odd, I found that either $p$ or $q$ must be $2$ in order for $pq+11$ to be a prime number. From here, I…
Silverleaf1
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