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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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Explanation why $\frac{(x-1)!-(x-1)}x$ is always an integer for prime $x$

If $x$ is prime, $\frac{(x-1)!-(x-1)}x$ is always an integer. Is there a way to explain this using modern algebra? I feel as if though it has to do with the concept of relatively prime or $\gcd$s, but I'm not exactly sure what is happening. I can…
rain
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How to go about proving that $p$ is prime?

Let $p\in\{2,3,4,\dots\}$. Suppose that $\forall x,y\in\mathbb Z\ p\mid xy\implies p\mid x\lor p\mid y$. Show that $p$ is prime. I am not fully understanding this problem. If I input numbers into the equation so for example $x = 5, y = 10$ and $p =…
fsdff
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distribution of prime powers

Let $f(x)$ be the number of prime powers less than $x$ (by prime power I mean any number of the form $p^n$ such that $p$ is prime and $n \ge 1$.) Consider the limit: lim x->∞ f(x)/x I would like to know the value of L In particular, this article…
Mathew
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If $2^x\bmod x = 2$, how does that prove that $x$, a prime number, is prime?

I saw this on l. 13 of this code here. I don't have a degree in maths but have read about Fermat's Theorem (which a comment below prompted me to remember), and have done first year university calculus and algebra, plus self-study of maths related to…
James Ray
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Proof of the infinitude of primes

Show that the integer $Q_n=n!+1$, where n is a positive integer, has a prime divisor greather han n. Conclude that there are infinity many primes. Answer is given by: Why is the following true: 1) $p|n!\Rightarrow p|(Q_n-n!)$ 2) We came true a…
WinstonCherf
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last digits for $p(n) + p(n+1)^2 + p(n+2)^3$

Let p(n) be the nth prime. Find sum $p(n) + p(n+1)^2 + p(n+2)^3$ and examine the last digits for their distribution. Taking the 64 possible patterns for last digits in these sums, one get 14 for ending digits 1 and 9, and 12 each for ending digits…
J. M. Bergot
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Show that $\text{rad}(n)=n$ when $n$ is a positive integer if and only if $n$ is square-free

Show that $\text{rad}(n)=n$ when $n$ is a positive integer if and only if $n$ is square-free. I came so far: $n$ is square free $\Rightarrow$ no $x^2$ divides $n \Rightarrow$ $n=p_1* ...*p_n\Rightarrow\text{rad}(n)=n=p_1* ...*p_n$.
WinstonCherf
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Two prime numbers' relationship

Let p1 and p2 be prime numbers and p1 < p2. Proove that there is a natural number n for which: p1 + n(p2 - p1) is prime AND p2 + n(p2 - p1) is not prime I've checked for n = 1, n = p1 and all of the answers that sounded reasonable. I tried also…
Billy Joel
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Longest prime containing primes

There are this interesting properties about prime numbers, where a number of $n$ digits can have $k$ primes within it's digits Examples: $1999$ has three primes in it, $\color{blue}{19}99, \color{blue}{199}9, \color{blue}{1999}$. Therefore we can…
Rab
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Prove that there is no non-constant polynomial $p(n)$ with integer coefficients that only takes prime values

Attempt (so far): Assume there exists a non-constant polynomial $p: \mathbb Z \to \mathbb Z$ with integer coeffecients that only takes on prime values. Let notate it as $$p(n)=d_j n^j + d_{j-1} n^{j-1}+ \dots +d_1 n + d_0 $$ where $j\in \mathbb…
Andrew Tawfeek
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For which $n>6$ is $11^{2^n} -2$ prime

Inspired by this answer to a cops-and-robbers code challenge The challenge was to create a program that gives one specific output for one single input number and another output for all others. The code checks that $11^{2^n}-2$ is prime and $n>6$,…
JollyJoker
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What are the different studies on prime numbers like Ulam's square?

I was writing a paper on prime numbers distribution. I am using graphical methods to show some observation about the nature of prime numbers. The article published by the Polish mathematician Stanislav Ulam is a mathematical note. Therefore my…
Vinay5forPrime
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How many numbers exists that are smaller than $p$ and prime with $p$?

I have a homework to hand in and they asked this question. I don't know if I'm supposed to count 1 as a prime to that number or not. In my case $p=3947$, so I count 3945 numbers fitting that criteria since $p$ is prime. Is this correct ? Thanks.
BrainOverfl0w
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Can you use 2^n - 2 / n to check if a number is prime with 100% accuracy?

According to the AKS primality test: $$(x-1)^p - (x^p-1)$$ If all coefficients (which can be found in Pascal's triangle) are divisible by p then p is prime. If we sum these coefficients we get: $2$ for $p = 2$; $6$ for $p = 3$; $14$ for $p =…
Tom
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prime number and modulus

I'm very curious about this cause I saw it done, but can't understand how and here it goes: PrimeNumber * SomeNumber % 26 Now if I know the prime number, and the 26 And the result, can I find the "SomeNumber"? Thanks Edit: I'm not good with math, so…
Not Amused
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