Prime numbers are natural numbers not divisible by any smaller number other than 1. This tag is intended for questions about, related to, or involving prime numbers.

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Q: Understanding Answer of 2012 AMC 8 - #18

The problem is: "What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?". The solution for this problem goes like this: "Since the integer ...
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Prove that 1 is the only “common” divisor of the integers n and n+2

Let n be any odd integer. Prove that 1 is the only "common" divisor of the integers n and n+2. I think you have to find gcd(n, n+2) and say that since n odd then then n+2 will also be odd. Thus n + ...
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Sets of Prime Numbers Generated By an Irreducible Monic Polynomial

Given a non-constant integral irreducible monic polynomial $f(x)$, the prime factors of its value at integers $x\in\mathbb{N}$ forms a set $\mathcal{P}(f)$. Is it possible that ...
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Prime number upper bound

I am reading some written notes about a proof I do not understand, maybe some informations are missing. The result that has to be proved is the following: if $p_n$ is the $n$-th prime number, ...
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Why isn't $1$ a superior highly composite number?

A superior highly composite number is a positive integer $n$ for which there is an $\epsilon>0$ such that $\dfrac{d(n)}{n^\epsilon} \geq \dfrac{d(k)}{k^\epsilon}$ for all $k>1$, where the ...
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How I could transform this into product over primes :$s_p$= $\frac{1}{2^2-1}+\frac{1}{2^3-1}+…\frac{1}{2^p-1}$?

1)Can I transforme this sum into product OVER primes:$s_p$= $\frac{1}{2^2-1}+\frac{1}{2^3-1}+....\frac{1}{2^p-1}$ ? Note : p is prime number and ${2^p-1}$ is prime 2)I would be interest to know ...
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What are the properties of a prime number?

For instance, we know that odd numbers behave like: $$x = 2y + 1 \quad\text{where}\quad x,y\in\mathbb Z$$ For even numbers: $$a = 2b \quad\text{where}\quad a,b\in\mathbb Z$$ But what about prime ...
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Why elements of the set can be Goldbach pairs for a given even number?

Let's take even number $100$ as an example (an example in the paper): Fom $2$ to $\sqrt{100}$ there's four primes:$\ 2,\ 3,\ 5,\ 7.\ $Let $$ \begin{align*} &A=\{n: n \in \mathbb{Z^+}, ...
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Largest $k$ such that $(p-k) = \lceil \sqrt{(p-2k) p} \rceil$

Assume $p \in \mathbb P.$ Assume $0<p-2k<p$ and the next square larger than $p(p-2k)$ is $(p-k)^2$. It is trivial to show that $p(p-2k)+k^2$ is a square. Simply $p(p-2k)+k^2 = (p-k)^2.$ ...
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Prime number question

Can somebody please give me a hint on how to start this question: Let $a$ and $n$ be two positive integers with $a,n ≥ 2$. Assume that $a^n−1$ is a prime number. Prove that $a = 2$ and $n$ is a prime ...
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Given $N$, what is the next prime $p$?

Certain data structures in programming related to collections operate in an optimal way if they have prime number of elements. This means if a program (programmer) requires $N$ (any natural number) ...
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What is the 5000th happy prime number?

Im writing a program that finds the Nth happy prime number. I think it works, but to double check I want to compare what it returns for the 5000th happy prime number. The problem is, I dont know where ...
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The $i^{th}$ prime in a given ring R

When I say that $p_1=2$, I mean that the first prime in the standard ring of integers $(\mathbb{Z},*,+)$ is $2$. I was wondering whether the notion of ordering the primes like this can be generalized ...
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Is this Goldbach-type problem easy to solve?

Problem: Given an odd prime number $p$, are there odd prime numbers $q$, $p'$, $q'$ such that $\{p,q\} \neq \{ p',q'\}$ and $p+q = p'+q'$ ? This comment informs that it's an obvious ...
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Bijection between Prime numbers and Natural numbers

We know that if set $S$ is countable then this set and set of all natural numbers are equivalent, which means that there must be some bijection between this two sets $F:S\rightarrow N$. We know that ...
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Is $2013^{2014}+2014^{2015}+2015^{2013}+1$ a prime? (usage of a computer not allowed)

Prove or disprove: $$2013^{2014}+2014^{2015}+2015^{2013}+1$$ is a prime number, without using a computer. I tried to transform the expression $n^{n+1}+(n+1)^{n+2}+(n+2)^{n}+1$, but couldn't reach ...
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Interchanging limits with the prime counting function

How does one justify that $$\lim_{s \to 1} \lim_{x \to \infty} \frac{\pi(x)}{x^s} = \lim_{x \to \infty} \lim_{s \to 1} \frac{\pi(x)}{x^s}, \quad s > 1,$$ without using the fact that the primes have ...
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Extending $f(p^k)$ where $p$ is prime

If we have a function $f(x)$, for which we know that $f(p^k)=(p^s+1)^k p^{sk}$ where $p$ is prime, $k$ is a real number, and $s$ is a constant, how do we find $f(x)$? My try: let $k=\log_p(x)$, so ...
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Given a prime p and an integer N, find the number of integers n such that 1≤n≤N and order(n!) is divisible by p

We are given a prime number $\leq 10^{18}$ and an integer N $(\leq N\leq 10^{18})$ how to find the number of integers lying in the range $1\leq n\leq N$ for which the order(n!) is a multiple of p? ...
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What is the mean prime power?

I found this definition in a book and I did not understand the meaning of it There exists a field of order q if and only if q is a prime power (i.e., $q = p^r$]) with p prime and r ∈ N. Moreover, if q ...
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Sum of reciprocals of primes for known primes.

I was reading through some old analytic number theory notes earlier and found the interesting fact that even though $\sum\frac{1}{p}$ diverges: $\sum_{\text{known primes}}\frac{1}{p} < 4$. ...
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Prime Partition

A prime partition of a number is a set of primes that sum to the number. For instance, {2 3 7} is a prime partition of $12$ because $2 + 3 + 7 = 12$. In fact, there ...
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Power of primes

We have proved that if $a > 3$, then $a$, $a+ 2$, and $a+ 4$ cannot be all primes in previous question. Can we say that they all be powers of primes?
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Extending primes

This question is more of a curiosity than anything. Start with a prime number and consider concatenating digits onto the right hand side. Sometimes you can make a prime and continue the process ...
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Prime factorization, distinct primes

Let $n=p^eq^f$ where $p$ and $q$ are distinct primes and $e$ and $f$ are positive integers. Show that $n$ has $(e + 1)(f + 1)$ distinct factors in $N$, and that the sum of all these factors is ...
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Sum of three primes equal to a prime [closed]

Does anyone know how to always get a prime from the sum of three primes? For example: 5+7+11=23, 17+29+43=89, etc.
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prime numbers and some conjectures

Consider triples $(p,q,r)$ of prime numbers $p$, $q$ and $r$ such that $(p+1)(q+1)=(r+1)$. Here are some examples : $(2,3,11), (3,7,31)$. How to prove there are infinitely many such triples?! I ...
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Primes Number Theory

For which primes $p$ is $2^p+1$ divisible by $p$? What I have been doing is: $2^p+1\equiv 0\pmod p$ $2^p\equiv -1\pmod p$ Then by Fermat's Theorem, we get $2^p\equiv 2\pmod p$ This shows ...
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A problem in prime number theory

I was wondering if anybody here might provide me with a hint for this rather innocuous-looking problem: If $X:= \{pq: p, q \mbox{ are prime numbers and } p\neq q\}.$ In addition, let us suppose that ...
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Prime numbers like 113

The number 113 is prime. The sum, product and all permutations of it's digits are prime. Are there any other such prime numbers?
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$n^2-79n+1601$ always a prime?

I am struggling with proving or disproving this: $n^2-79n+1601$ is a prime for all natural numbers $n$ (except multiples of $1601$). This somehow has a relation to Stanislaw Ulam spiral. What ...
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How does Hildebrands proof of the prime number theorem via large sieve work?

How does the sieve inequality (I may not know the most general form) lead to the distribution of primes? To me, these concepts do not seem to be related. Can their connection be described in a ...
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How to prove that this Proth number cannot be a prime number? (without using a computer)

Without using a computer prove that this Proth number cannot be a prime number : $$43373\cdot 2^{49822}+1$$
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Congruences with prime number and factorial

Prove that if $p\equiv 1 \pmod{4}$ is a prime number and $$x\equiv \pm \left(\frac{p-1}{2}\right)! \pmod{p}$$ then $x^2\equiv -1 \pmod{p}$ I think Wilson's theorem will come in handy here, used ...
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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 ...
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Prove or disprove: $99^{100}+100^{101}+101^{99}+1$ is a prime number

Prove or disprove: $$99^{100}+100^{101}+101^{99}+1$$ is a prime number. My idea: let $100^{101}=x^{x+1}$,then $$99^{100}+100^{101}+101^{99}+1=(x-1)^{x}+x^{x+1}+(x+1)^{x-1}+1$$ is prime number? I ...
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Related to greatest prime number that divides $n.$

I don't even have an idea of how to start working on this one: Let $p(n)$ denote the greatest prime number that divides $n.$ Show that there exists infinitely many positive integers $m$ such that ...
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How to prove a number is not a prime number (without a computer)

Show that $$5994937829$$ is not prime number How can I use math methods to prove it, and I know that this be proven using computer. But I can use only math methods to solve it.
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What's the best software for primality tests of huge numbers? (check if an integer is prime or not)

I just read an article about huge prime numbers (some with more than 10millions digits!) that are discovered using software that check if an integer is prime or not (primality test sofwares). What is ...
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Density in $\mathbb{R}_{ +}$ of a subgroup of $\mathbb{Q}_{> 0}$?

Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number. It follows that $\phi(1)=0$, $\phi(a.b) = \phi(a)+\phi(b)$, $\phi(a^{-1}) = ...
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Do 4 consecutive primes always form a polygon?

Related to this question, if 4 segments have length of 4 consecutive primes, can they always form a 4-vertex polygon? This question occurred to me out of sheer curiosity, but now I can't prove or ...
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Is there a prime between $k$ and $\dfrac{11}{9}k$, $\forall k\ge 24$?

Given $k\in\mathbb{N}$, $k\ge 24$, is there always a prime number in the interval $\left[k,\dfrac{11}{9}k\right]$? I tried to verify this statement with the computer and it seems to hold. Is it ...
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An integral and $\pi(n)$

Are there polynomials $P,Q\in \mathbb{R}[x]$ satisfying : $$\int_{0}^{\log n}\frac{P(x)}{Q(x)}\,\mathrm{d}x=\frac{n}{\pi(n)}\quad \text{ for infinitely many }n\in \mathbb{N}$$ Here $\pi(n)$ is the ...
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On a topological proof of the infinitude of prime numbers.

There is a proof of the infiniteness of prime numbers using Topology. I was only informed of the existence of this proof. They say it's very elegant. One could show how this proof?
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Infinite families of prime numbers

What interesting/useful infinite families of prime numbers are there? Right now it would be useful if I could find one with arbitrarily many 1's in its binary representation, but I am doing a larger ...
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Math Olympiad Prime Number Question

If $p$, $q$ and $r$ are prime numbers such that their product is $19$ times their sum, find $p^2$ + $q^2$ + $r^2$. I came across this question in a Math Olympiad Competition and had no idea how ...
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Prime powers, patterns similar to $\lbrace 0,1,0,2,0,1,0,3\ldots \rbrace$ and formulas for $\sigma_k(n)$

Some time ago when decomponsing the natural numbers, $\mathbb{N}$, in prime powes I noticed a pattern in their powers. Taking, for example, the numbers $\lbrace 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16 ...
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Proof that $123456789098765432111$ is prime?

The mathematician Charles Weibel asks on his home page the following "fun question": How can you prove that 123456789098765432111 is a prime number? (He notes the fact $$12345678987654321 = ...
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Last digits of primes

A prime number not equal to $2$ and $5$ can't have last digit equal to $2,4,5,6$ and $8$. Is it true that this is the only restriction on last digits of prime numbers? I mean if its true that for any ...
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Characterizing the primes which don't divide any Pell-Lucas number(s)

For integer $n$, let $P_n$ be a Pell number, and $Q_n$ its companion. Is there a characterization of the prime numbers $p$ which don't divide any $Q_n$? By brute-force search, I found that this ...