Mathematics Stack Exchange
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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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Curiosity About the Infinitude of Primes via Concatenation of Distinct Primes

Question: Do there exist infinitely many primes of the form $p*q,$ where $*$ denotes concatenation and $p$ and $q$ are both prime? On my way into the grocery store the other day, I had the above curiosity pop into my head. Originally, I thought…
Dylan C. Beck
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Why must a and p be relatively prime in Fermat's Little Theorem?

A variant of Fermat's Little Theorem states that $a^{p - 1} \equiv 1~mod~p$ if $a$ is not divisible by $p$. Why is this last condition important? Why must $a$ and $p$ be relatively prime?
John Hoffman
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A mysterious prime number 127

$127$ is probably the LAST / LARGEST prime number $p$ such that $p^2\pmod{q}$ has an odd residue, where $q$ is the previous prime number right before $p$. I have checked it up to $10^6$ , and it turned out to have been checked up to $4\cdot10^{18}$…
Larry Zoo
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Let $m^n-1$ be prime. What can $m$ be?

Let $m^n-1$ be prime. What can $m$ be if $m$ and $n$ are not $1$? How can I find $m$?
user75283
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When GIMPS detects a composite number, does it continue with some calculations until the estimated completion date?

I joined GIMPS for a while and have identified several composite numbers. For those numbers assigned to me for PRP tests, the estimated completion dates and the number of required iterations are always very accurate. How can the estimation be so…
modnar
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how to prove conjecture $\gcd(p^2+3p+4,2^p-1)=1$

if the prime number such $p\equiv 5\pmod 8,p\neq 29$,I conjecture $$\gcd(p^2+3p+4,2^p-1)=1,p\neq 29$$ I test $p<100$ is true
math110
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three consecutive numbers with exactly different four prime factors

The three consecutive numbers 127, 128, 129 have exactly four different prime factors, namely, 2, 3, 43, and 127. Are these numbers infinite?
J. M. Bergot
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Primes of the form $a^n-(a-1)^n$

For a given $n$, consider the assertion: $\exists a \in \Bbb Z : a^n-(a-1)^n\ \text{is prime}\tag*{}$ How can one do one of the following: Prove that the assertion is true for all integer $n > 1$ Prove that the assertion is false for all integer $n…
Ted Hopp
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Counting zeros and ones in binary representations of prime numbers

If you write down binary representations of all prime numbers starting from 3 up to some (very big) $N^{th}$ prime number and denote with $S_1(N)$ the total number of ones (1) and with $S_0(N)$ the total number of zeros (0) used in all binary…
Saša
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Use Fermat's Theorem to prove 10001 is composite

I need to use Fermat's Theorem to prove that 10001 is not prime. I understand that I just need to find a counterexample where $a^{10000}$ mod 10001 = 1 mod 10001 does not hold true, but this seems kind of difficult with such large numbers. Any…
katie
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Let $t_1$, $t_2$,.... $t_n$ is a sequence where $t_1=2$ and $t_{n+1}=t_n{^2}-t_n+1$. Prove that if $m\ne n$, then $t_m$ and $t_n$ are coprime.

Let $t_1$, $t_2$,$\enspace$.... $t_n$ is a sequence of natural numbers. The sequence is defined by these equalities - $t_1=2$ $\enspace$ and $\enspace$ $t_{n+1}=t_n{^2}-t_n+1$. $\enspace$ Prove that if $m\ne n$, $\enspace$ then $t_m$ and $t_n$…
john1672
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Root 2 primes make up 38% of all prime numbers.

It has been conjectured by Emile Artin that the primes of primitive root 2 are infinite. Is it known that their density among all primes is about 1/3 and that this density is constant up to 1 quadrillion? I have a quick test for determining if…
Casey Stewart
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Why don’t mathematicians work on ‘difference-asymptotic’ of prime counting function?

Mathematicians back in 19th century tried to find a function that satisfies $$\lim_{x\to\infty}\frac{\pi(x)}{f(x)}=1$$ and $f(x)$ turns out to be $\frac{x}{\ln x}$, or any function asymptotic to it(like $\text{Li}(x)$). They proved it rigorously and…
Szeto
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Asymptotic expression for the $n$th prime number

Quoting from the Wikipedia article: As a consequence of the prime number theorem, one gets an asymptotic expression for the $ n $th prime number, denoted by $ p_n $: $$ p_n \sim n \log n.$$ Can you explain how we get this approximate expression? I…
Lone Learner
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show this number maybe is prime?

prove or disprove $$2016^{2017}+1008^{2017}\cdot 2017^{1008}+(2017)^{2016}$$ not prime number? It's probably based on factorization. $2016=1008\cdot 2$
math110
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