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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The n-th prime is less than $n^2$?

Let $p_n$ be the n-th prime number, e.g. $p_1=2,p_2=3,p_3=5$. How do I show that for all $n>1$, $p_n<n^2$?
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Proof of infinitude of primes using the irrationality of π

According to the section Proof using the irrationality of $\pi$ of the Wikipedia article on Euclid's theorem, Euler proved that: ...
15
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1answer
387 views

Primey Pascal's Triangle

Imagine that we have a triangle that starts with 2,3 and grows like Pascal's triangle but instead uses the smallest prime $\geq$ to the sum of the above two primes. Visually: $$ ...
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why $ \rm{lcm}[1,2,3,\cdots,n]\in (2^n,4^n)$

Let $n\ge 7$ be positive integers,show that $$f(n)=\rm{lcm}[1,2,3,\cdots,n]\in (2^n,4^n)$$ Anyone know this problem background?or maybe have best proof or best result?
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If the abc conjecture has been proven what implication does that have for elliptic curve cryptography?

I am not a mathematician, but I was wondering if the proposed proof of the abc conjecture (PDF) by Shinichi Mochizuki of Kyoto University would contain insights and mathematical tools that would lead ...
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Quadratic sieve algorithm

I am stuck with the sieving stage of Quadratic Sieve algorithm. I've read lots of papers to this point but I can't find any guidlines how to choose sieving interval or how sieving is actually done ...
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Algorithm to find primes up to $n$ in $O\left(\frac{n}{\log n}\right)$?

Consider the problem of given an integer $n$, generating a list of the primes not greater than $n$. An optimized version of the Sieve of Eratosthenes can do such task in $O(n)$, while the more modern ...
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Is an arbitrary number of the form xyzxyz divisible by 7, 11, 13?

So I was given this question Choose any 3-digit number xyz and write it after itself as follows: xyzxyz. Check whether it is divisible by 7,11, 13. Is an arbitrary number of the form xyzxyz ...
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Determine whether a number is prime

How do I determine if a number is prime? I'm writing a program where a user inputs any integer and from that the program determines whether the number is prime, but how do I go about that?
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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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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 ...
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Finding smallest and largest prime factor of $\frac{200!}{180!}$

I originally started off by listing all the primes: $p<200$ then trying to calculate the prime factorisation of each (which I realise is a silly thing to do) I believe there must be a simpler way ...
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$\sum\limits_{\text{prime }p} 2^{-p}$ is an irrational number

I need help to prove the following result. $\displaystyle\sum_{\text{prime }p} 2^{-p}$ is an irrational number.
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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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4answers
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Why do repunit primes have only a prime number of consecutive $1$s?

Repunit primes are primes of the form $\frac{10^n - 1}{9} = 1111\dots11 \space (n-1 \space ones)$. Each repunit prime is denoted by $R_i$, where $i$ is the number of consecutive $1$s it has. So far, ...
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2answers
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Evidence against Goldbach's Conjecture?

It recently occurred to me that, unless I'm much mistaken, Goldbach's conjecture can easily be seen to be equivalent to a seemingly more general statement: Every number $n$ divisible by any ...
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Are the logarithms in number theory natural?

I find the frequent emergence of logarithms and even nested logarithms in number theory, especially the prime number counting business, somewhat unsettling. What is the reason for them? Has it maybe ...
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German sofa primes: Can both $q$ and $\frac{q^3+1}{2}$ be prime?

Is there an odd prime integer $\displaystyle q$ such that $\displaystyle p= \frac{q^3+1}{2}$ is also prime? A quick search did not find any, nor a pattern in the prime factorization of p. This ...
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2answers
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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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1answer
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Prime with digits reversed is prime?

Well, just another idea came up into my mind and i have no idea how to solve it :D Is there infinitely many prime numbers, which are not repunits and their inverse is also prime? (For example, inverse ...
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2answers
391 views

Numbers that are divisible by the number of primes smaller than them

Let $\pi(n)$ denote the number of primes less than or equal to $n$ (a.k.a the prime-counting function). For certain values of $n$, the value of $\frac{n}{\pi(n)}$ is integer. Here are the first few ...
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sums of square free numbers , is this conjecture equivalent to goldbach's conjecture?

As one can notice every integer greater than $1$ is a sum of two squarefree numbers.(numbers that are not divided by some prime square power). Can we prove that? Edit: Can we have bounds for the ...
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3answers
359 views

If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.

The proof is already given in the textbook but I tried other way around. Proof by contradiction: Let's assume that $p$ doesn't divide $a$ and $p$ doesn't divide $b$, but $p$ divides $ab$. So ...
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Proof involving induction and primes

I'm looking to prove that: $$p_n \leq 2^{2^{n-1}}$$ Where $p_n$ denotes the $n$th prime in ascending order. The proof method is induction. I've solved my base case, that is: $n=1$ $p_1 = 2$, ...
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Primes sum ratio

Let $$G(n)=\begin{cases}1 &\text{if }n \text{ is a prime }\equiv 3\bmod17\\0&\text{otherwise}\end{cases}$$ And let $$P(n)=\begin{cases}1 &\text{if }n \text{ is a prime ...
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A conjecture concerning primes and algebra

A monoid morphism $\psi:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ is defined by an arbitrary function $f:\mathbb Z_+\!\!\rightarrow\mathbb Z_+$ and defines a group homomorphism $\varphi:\mathbb ...
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1answer
280 views

Prime pair points slope approaches 1

Take the list of primes, $$2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, \ldots$$ and create ordered $(x,y)$ pairs by grouping in sequence, $$(2, 3), (5, 7), (11, 13), ...
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Sum of reciprocals of primes factorial: $\sum_{p\;\text{prime}}\frac{1}{p!}$

The series $$\sum_{p\;\text{prime}}\frac{1}{p}=\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\cdots$$ diverges as is well known. How about the following? ...
14
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1answer
586 views

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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Would proof of Legendre's conjecture also prove Riemann's hypothesis?

Legendre's conjecture is that there exists a prime number between $n^2$ and $(n+1)^2$. This has been shown to be very likely using computers, but this is merely a heuristic. I have read that if this ...
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Twin, cousin, sexy, … primes

Twin, cousin, and sexy primes are of the forms $(p,p+2)$, $(p,p+4)$, $(p,p+6)$ respectively, for $p$ a prime. The Wikipedia article on cousin primes says that, "It follows from the first ...
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1answer
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Are all primes Euclid primes?

I was learning Euclid's theorem. If we repeat his construction (properly modified to give only primes) then will we skip any primes? Formally: $p_1 = 2$, and $p_{n + 1} = \text{smallest} \; q \in ...
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1answer
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Simple(r) proof that $\pi(2^n)\geq n$?

We can clearly prove that $\pi(2^n)\geq n$ with Bertrand's postulate, but that seems like overkill. Is there any simpler way one can prove that $\pi(2^n)\geq n$? Note: $\pi(m)$ is the prime ...
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1answer
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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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202 views

Consequences of the negation of the Riemann hypothesis

There are many sources documenting the consequences of the Riemann hypothesis, but I can't find one discussing the consequences of its negation, particularly concerning the prime distribution. Can ...
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211 views

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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Going from $\Lambda$ to a prime count

A 1997 paper of Étienne Fouvry and Henryk Iwaniec, Gaussian primes, concerns the prevalence of primes that are of the form $n^2+p^2$ for prime $p$. The asymptotic result is $$\sum_{n^2+p^2\le ...
14
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Primes as quotients

I ask this question based on a comment of David Speyer in another question. What primes are of the form $$ \frac{p^2-1}{q^2-1} $$ where $p$ and $q$ are prime? The first prime not apparently of this ...
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Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
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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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Why does one counterexample disprove a conjecture?

Can't a conjecture be correct about most solutions except maybe a family of solutions? For example, a few centuries ago it was widely believed that $2^{2^n}+1$ is a prime number for any $n$ . For ...
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Does the formula $\sqrt{ 1 + 24n }$ always yield prime?

I did some experiments, using C++, investigating the values of $\sqrt{1+24n}$. ...
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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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What is the symbol for primes?

Although there isn't much difference between $\mathbb{Z},\mathbb{N},\mathbb{I}$, they are well-known, and each one gets its own distinguished symbol. Is there any reason that primes don't get their ...
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Can exist an even number greater than $36$ with more even divisors than $36$, all of them being a prime$-1$?

I did a little test today looking for all the numbers such as their even divisors are exactly all of them a prime number minus 1, to verify possible properties of them. These are the first terms, it ...
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1answer
283 views

What is the smallest integer $n$>1 such that $n^{5000}+n^{2013}+1$ is prime?

Which is the smallest integer $n>1$, such that $$n^{5000}+n^{2013}+1$$ is prime ? Since $x^{5000}+x^{2013}+1$ is irreducible over $\mathbb{Q}$ and has value $1$ for $x=0$, there should be ...
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1answer
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Chebyshev: Proof $\prod \limits_{p \leq 2k}{\;} p > 2^k$

How do I prove the following: $$\prod_{p \leq 2k} \; p > 2^k \text{ with } p \in \mathbb{P}$$ I tried induction, but I didn't know how to go on because I don't have a look at all numbers. ...
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$x^2+x+1$ is the cube of a prime.

Please help me find all natural numbers $x$ so that $x^2+x+1$ is the cube of a prime number.(Used in here)
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Ulam spiral: Is there an “unusual amount of clumping” in prime-rich quadratic polynomials?

I was reading Martin Gardner's Mathematical Games column on the Ulam spiral which appeared in the March 1964 issue of Scientific American. (The spiral actually featured on the cover of that issue.) ...