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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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Show there is no prime in a range of numbers

How do I show that except for 5039, there is no prime between 5033 and 5047. I just need a nudge in the right direction, no idea how to start the problem :(
bob
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If $p \mid m^p+n^p$ prove $p^2 \mid m^p+n^p$

Prove that for a prime $p>2$, if $p \mid m^p+n^p$, prove $p^2\mid m^p+n^p$ From Fermats theorem I concluded $p \mid m+n$, so $p^2\mid (m+n)^p$. How do I proceed next? Any hints are welcomed.
Morty
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Arithmetic progression of primes question

Is it known whether for all positive integers $k$ there is an integer $a$ such that $a+30n$ is a prime number for all $n\in \{1,\ldots,k\}$?
Wondering
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Formula for likely prime

Numbers of the form $n!+1$ are quite often prime numbers. Is there any formula $f(n)$ such that the probability that $f(n)$ is prime approaches 1 as $n$ goes to infinity and $f(n)$ also approaches infinity? If not, is there some formula better then…
TROLLHUNTER
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Does factorization end with a prime number?

When doing factorization, I have always taught kids to work from the outside in. So for the number $28$, you start with $1$ and $28$, then $2$ and $14$, then $4$ and $7$. And once you reach the middle, you are finished. Because $5$, $6$, and $7$,…
Ken Lemos
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Let $x$ be greater than $1$. Prove $x$ is prime if and only if for every integer $y$, either $\gcd(x,y)=1$ or $x\mid y$.

I've been having serious trouble with this problem, The first direction-> Proving x is prime if for every integer y, either gcd(x,y)=1 or x|y doesn't seem too difficult. We know that if gcd(x,y)=1 then they are coprime.. but what does the fact that…
user122661
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The Number of Prime Factors

x equals any whole number. y equals the number of prime factors of x. You plot those points, then find a line of best fit. What would the equation for that line be? Also; why? $x = 48$ $y = 5$ Because $48 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3$
Peter
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How to prove a given number is prime?

How would I go about showing a number is prime, especially a very large number. Say I wanted to show that 43112621 is a prime number. How would I go about doing this without showing no other prime number prior to it?
Dexter Peters
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Counting primes by counting numbers of the form $6k \pm 1$ which are not prime

Again, pondering on twin primes, I came upon the following result. It baffles me a bit, so could someone give more intuitive reasoning why it works. First, define a function $P_6$ as $$P_6(n)=\begin{cases} 0, \ \ 6n-1 \not\in \mathbb P \wedge 6n+1…
Valtteri
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A curious plot of Primes (II)

Here is the plot of a function $f(x)$ such that: $$ f(x) = \frac {P_ {\lfloor 2 x \rfloor}} {P_ {\lfloor 2 x \rfloor - 1}} $$ where $P_k$ is the $k^{\mathrm{th}}$ Prime Number for x in range [1,300]. The curves are in the form $1 + \frac {1} {a…
Eddy Khemiri
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Why do we use Φ while calculating the encryption key in RSA method?

As you know, in RSA encryption, we calculate an n value by multiplying two long primes. Then we calculate Φ(n), which is equal to (p-1)(q-1). After that, we choose an encryption key which satisfies gcd(e,Φ(n))=1. I know that it is necessary to…
user13791
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Does euclid's proof of infinitude of primes generate all the primes if you keep adding the new primes to the list?

If I start with a set of consecutive primes, let's say $\{2,3,5,7\}$. Consider $2\times 3\times5\times7+ 1 = 211$. Now, $211$ is prime, so I add it to the set: $\{2,3,5,7,211\}$. If the resulting number is not prime, add the prime divisors of this…
Tora
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Finding prime number solutions of an equation

How can we find the prime numbers $p, q, r$ such that $5pqr-2p-10r = 270$? By using the fact that $$5pqr-2p-10r -270\equiv -2p\equiv 0\pmod{5}$$ $$p\equiv 0\pmod{5}$$ and that $p$ is a prime number, we impose that $p = 5$. Similarly, $$5pqr-2p-10r…
user1189470
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Is there a pattern of grouping for prime numbers?

Consider all the natural numbers up to 1000 ending in the number n, which has a value in the set (1, 3, 7, 9). Calculate n^2-2 for each individual value of n up to 1000 and check if the result is prime. Save the set of these prime results for n=1,…
Betho's
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I wrote a formula for prime numbers. Is it good enough to write article about it or not worth it?

Let $x\ge0$ and $x\in\Bbb Z$. I wrote function $a(x)$ that for $x\ge1$ returns prime $\ge7$ or composite of these primes, and for $x=0$ it returns 5 $$a(x) = \frac{9+6(x+\lfloor\frac{\vert…
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