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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Irrationality of “primes coded in binary”

For fun, I have been considering the number $$ \ell := \sum_{p} \frac{1}{2^p} $$ It is clear that the sum converges and hence $\ell$ is finite. $\ell$ also has the binary expansion $$ \ell = ...
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Prove a number is composite

How can I prove that $$n^4 + 4$$ is composite for all $n > 5$? This problem looked very simple, but I took 6 hours and ended up with nothing :(. I broke it into cases base on quotient remainder ...
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A prime of the form $38111111\ldots$

Let $z(n)$ denote the number given by $38$ followed by $n 1$'s. What is the least number $n$, such that $z(n)$ is prime ? With brute force, I checked up to $7000$ digits and did not find a prime. ...
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Is a prime factor of a number always less than its square root?

I was going through the fundamental theorem in Number Theory where any non zero integer n can be represented as a product of distinct primes. A related problem with this theorem is to prove that for ...
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Prove that $n^2+n+41$ is prime for $n<40$

Here's a problem that showed up on an exam I took, I'm interested in seeing if there are other ways to approach it. Let $n\in\{0,1,...,39\}$. Prove that $n^2+n+41$ is prime. I shall provide my own ...
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How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?

I saw reference to this result of Chebyshev's: $$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$ and its relation to the Prime Number Theorem. I'm looking into an ...
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Cardinality of the set of prime numbers

It was proved by Euclid that there are infinitely many primes. But what is the cardinality of the set of prime numbers ? Cantor showed that the sets $\mathbb{Q}$ and $\mathbb{Z}$ have the same ...
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What is the significance of the power of $3$ in the sequence of primes given by $\lfloor A^{3^n}\rfloor ?$

Mill's constant is a number such that $\lfloor A^{3^n}\rfloor$ is prime for all $n$. The existence of such an $A$ was proven in $1947$. I know little about number theory, but I am curious as to why ...
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How rare are the primes $p$ such that $p$ divides the sum of all primes less than $p$?

This is just for fun! The title pretty much says it all. It's probably a very difficult question. Up to the $40,000^{th}$ prime $(479909)$, I have found only $5$, $71$ and $369119$ with this ...
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Can you make any number prime by adding some digits to the right?

To ask in another way: Is it guaranteed that any given sequence of digits will be at the beginning of some prime number? We know from Dirichlet's theorem that there are an infinite number of values ...
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An upper bound for $\log \operatorname{rad}(n!)$

Let $n>1$ be an integer and let $\operatorname{rad}(n!)$ denote the radical of $n$-factorial. (The radical of an integer $m$ being, loosely speaking, the product of the prime divisors of $m$.) Can ...
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Find all primes $p$ such that $\dfrac{(2^{p-1}-1)}{p}$ is a perfect square

Find all primes $p$ such that $\dfrac{(2^{p-1}-1)}{p}$ is a perfect square. I tried brute-force method and tried to find some pattern. I got $p=3,7$ as solutions . Apart from these I have tried for ...
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Why are conjectures about the primes so hard to prove?

I recently started learning number theory, and I've noticed there are many conjectures about the prime numbers that are unproven. Some examples would be whether there are infinite Mersenne, ...
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Asymptotics of LCM

Let $\operatorname{LCM}(x_1,x_2,\ldots,x_n)$ be the least common multiple of the integers $x_i$. How can one find the asymptotics of $\operatorname{LCM}(f(1),f(2),\dots,f(n))$ as $n$ approaches ...
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How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis if at all?

How would proving or disproving the Twin Prime Conjecture affect proving or disproving the Riemann Hypothesis? What are the connections between both conjectures if any?
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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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crazy problem - does it have a solution? number theory perhaps?

A few researchers are trying to crack a code which involves discovering the values of three integers. They know they are between 1 and 100 (inclusive), and that they may be the same. They each have a ...
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The Goldbach Conjecture and Hardy-Littlewood Asymptotic

A source I am reading refers to the Goldbach conjecture (that every even number is the sum of two primes), and then immediately follows with the "Hardy-Littlewood conjecture" that $\sum ...
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What's the probability that a sum of dice is prime?

Prompted by today's Minute Math question on the MAA site (http://amc.maa.org/mathclub/5-0,problems/T-problems/T-web,ia/2005web/tb05-12-ia.shtml), I started thinking about the probability that the sum ...
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Are primes randomly distributed?

There is a famous citation that says "It is evident that the primes are randomly distributed but, unfortunately, we don't know what 'random' means." R. C. Vaughan (February 1990) I have this very ...
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Is there any theoretical indication that this conjecture of Catalan could be true?

Belgian mathematician Catalan in $1876$ made next conjecture: If we consider the following sequence of Mersenne prime numbers: $2^2-1=3 , 2^3-1=7 , 2^7-1=127 , 2^{127}-1$ then $$2^{2^{127}-1}-1$$ is ...
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Primes in $\lfloor a^{n} \rfloor$

Motivated by the question Is there any result, that says that $\lfloor e^{n} \rfloor$ is never a prime for $n>2$?, take a real number $a>1$ and consider the sequence $\lfloor a^{n} \rfloor$. ...
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Divergence of the Derivative of the Prime Counting Function

On the one hand, the Prime Counting Function $\pi_0(x)$ maybe be written $$ \pi_0(x) = \operatorname{R}(x^1) - \sum_{\rho}\operatorname{R}(x^{\rho}) \tag{1} $$ with $ \operatorname{R}(z) = ...
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Decimal form of irrational numbers

In the decimal form of an irrational number like: $$\pi=3.141592653589\ldots$$ Do we have all the numbers from $0$ to $9$. I verified $\pi$ and all the numbers are there. Is this true in general for ...
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Can true randomness come out of mathematical rules?

For example, prime numbers, they seem very random, and they are defined by a simple set of rules. I can't see how real randomness could exist in the real world, but what about mathematics?
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Disproving the claim that the numbers 1+2+4, 1+2+4+8, 1+2+4+8+16… alternate between prime and composite

I am working through an elementary number theory book and I have come across the following problem. Show the following claims are wrong: Claim 1: The sequence 1+2+4, 1+2+4+8, 1+2+4+8+16, ...
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If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$

If $n = 51! +1$, Then find no of primes among $n+1,n+2,\ldots, n+50$ Really speaking, I don't have any clue ...
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Why does “the probability of a random natural number being prime” make no sense?

I read an essay about prime numbers. In it the author suggests that a natural first question to ask is, What is the probability of a random natural number being prime? but proceeds to dismiss ...
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Are there any Combinatoric proofs of Bertrand's postulate?

I feel like there must exist a combinatoric proof of a theorem like: There is a prime between $n$ and $2n$, or $p$ and $p^2$ or anything similar to this stronger than there is a prime between $p$ and ...
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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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A Poster About Prime Numbers [closed]

We're going to design a poster about prime numbers, which will appear in a mathematics magazine for middle school students. The poster should be both visually attractive and mathematically rich. Do ...
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Why are all non-prime numbers divisible by a prime number?

In Euclid's infinite prime numbers proof, the logic is as follows: Assume a set $S$ of all prime numbers in existence is finite (there are a finite amount of primes) Then there must be a greatest ...
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Where is the fallacy in the argument using Prime Number Theorem

I am reading about Prime Number Theorem from book by Ingham. As as application of PNT I found the following theorem: Now my doubt is at the step $\frac{\log(y)}{\log(x)}\rightarrow 1$, we can say ...
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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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prove that $\lim_{x\to\infty} \pi(x)/x=0$

I think I might have asked this question before, but I can't find it on the site, so I sincerely apologize if I am making a duplicate. But anyway, I have been working on this proof for several weeks ...
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Distribution of primes?

Do primes become more or less frequent as you go further out on the number line? That is, are there more or fewer primes between $1$ and $1,000,000$ than between $1,000,000$ and $2,000,000$? A proof ...
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Consecutive Prime Gap Sum (Amateur)

List of the first fifty prime gaps: 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4. My ...
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Take any number and keep appending 1's to the right of it. Are there an infinite number of primes in this sequence?

Ignoring sequences that are always factorable such as starting with 11, Can we take any other number such as 42 and continually append 1s (forming the sequence {42, 421, 4211, ...}) to get a sequence ...
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Mapping natural numbers into prime-exponents space

Take any natural number $n$, and factor it as $n=2^{e_1} 3^{e_2} 5^{e_3} ... p^{e_i}$, where $i$ is the $i$-th prime. Now map $n$ to the point $n \mapsto (e_1,e_2,\ldots,e_i,0,\ldots)$, where $i$ is ...
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A problem about the largest prime factor of $n^2+1$

Let $f(n)$ be the largest prime factor of $n$. The image of function $g(n)=\sqrt{f(n^2+1)}$ is like this: Question: If we want to draw a horizontal line which bisects the points from $n=1$ to ...
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How to find the solutions for the n-th root of unity in modular arithmetic?

$$\begin{align*} x^n\equiv1&\pmod p\quad(1)\\ x^n\equiv-1&\pmod p\quad(2)\end{align*}$$ Where $n\in\mathbb{N}$,$\quad p\in\text{Primes}$ and $x\in \{0,1,2\dots,p-1\}$. How we can find the ...
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What is the intuitive meaning of “conspiracy” in number theory?

Assuming very little number-theoretic background from my part, could you please explain me what is the intuitive meaning behind "conspiracy" in number theory? There is no formal entry on Wikipedia and ...
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Is the set of all numbers which divide a specific function of their prime factors, infinite?

Consider a number $n$ with prime factorization $n=p_1^{k_1} \cdot p_2^{k_2} \dots \cdot p_z^{k_z}$. We define a function $f(n)$ to be $f(n)=(p_1^{k_1+1}-1) \cdot (p_2 ^{k_2 +1}-1) \dots \cdot ...
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Probability p+k is a prime

If p is a prime number, and k is an even integer, what is the probability p+k is a prime number? According to my simulations p+108 is prime twice as often as p+344
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Is there a single or best reason that 2 is an exceptional prime?

I've recently been studying some elementary number theory, and I've frequently come across the fact that there are a fair number of results (the main one being the law of quadratic reciprocity) for ...
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Primes of the form 1..1

For $n \ge 1$ an integer, let's denote $u_n = \sum_{k = 0}^{n-1} 10^k$ That is $u_1 = 1$, $u_2 = 11$, $u_3 = 111$, $u_4 = 1111$, ... My question is the following : Which of them are prime numbers ? ...
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About prime factor and consecutive integers

The problem is: There exists an integer $N$ such that for any $n>N$, there exists $m \in \{n,n+1, \ldots ,n+9\}$ such that $m$ has at least $3$ distinct prime factors. 2 Years ago, My ...
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Are closed geodesics the prime numbers of Riemannian manifolds?

I wonder to what extent one can support the analogy that primitive closed geodesics are the prime numbers of Riemannian manifolds? ("Primitive": traced once, as opposed to $m$-fold for $m \ge 2$.) In ...
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Can the order of 2 mod p be arbitrarily small (relative to $p - 1$)?

Given a prime number $p$, let $\operatorname{ord}_p(2)$ be the multiplicative order of $2$ modulo $p$, i.e., the smallest integer $k$ such that $p$ divides $2^k - 1$. By Lagrange's theorem, ...
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The Fermat prime 257 and binomial sum $\sum_{n=0}^\infty \frac{(-1)^n}{\binom {8n}{4n}}$?

We have, $\begin{aligned} \sum_{n=0}^\infty \frac{(-1)^n}{\binom n{n/2}} &= \frac{4}{27}(9-\pi\sqrt{3}\,)\\[2.5mm] \sum_{n=0}^\infty \frac{(-1)^n}{\binom {2n}n} &= \frac{4}{5} - ...