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.

learn more… | top users | synonyms

11
votes
3answers
1k views

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 ...
11
votes
4answers
946 views

Proof that there are infinitely many prime numbers starting with a given digit string

To prove the following fact: given any sequence of digits in any base, eg 314159265358979323 base 10, there are infinitely many primes that start with these digits,eg when expressed in decimal they ...
11
votes
1answer
143 views

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 ...
11
votes
2answers
338 views

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 ...
11
votes
2answers
569 views

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 ...
11
votes
2answers
221 views

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 ...
11
votes
1answer
450 views

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 ...
11
votes
3answers
320 views

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, ...
11
votes
2answers
419 views

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 ...
11
votes
2answers
642 views

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?
11
votes
3answers
250 views

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 ...
11
votes
1answer
1k views

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 ...
11
votes
2answers
751 views

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 ...
11
votes
1answer
372 views

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 ...
11
votes
1answer
686 views

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 ...
11
votes
2answers
617 views

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 ...
11
votes
1answer
287 views

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 ...
11
votes
0answers
135 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 $\lfloor a^{n} \rfloor$. ...
11
votes
0answers
324 views

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) = ...
10
votes
5answers
933 views

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 ...
10
votes
7answers
600 views

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?
10
votes
6answers
611 views

Define an infinite subset of primes such that the sum of reciprocals converges

How can we define an infinite subset of primes such that the sum of reciprocals converges? $S=\{p\in \mathbb{Z}^+ : p\ \text{is prime and some condition on}\ p\}$ s.t. ...
10
votes
5answers
2k views

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, ...
10
votes
4answers
844 views

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 ...
10
votes
3answers
340 views

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 ...
10
votes
4answers
1k views

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 ...
10
votes
9answers
932 views

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 ...
10
votes
3answers
2k views

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 ...
10
votes
3answers
227 views

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 ...
10
votes
2answers
1k views

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 ...
10
votes
3answers
506 views

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 ...
10
votes
2answers
366 views

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 ...
10
votes
3answers
320 views

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 ...
10
votes
2answers
306 views

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 ...
10
votes
2answers
567 views

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 ...
10
votes
2answers
435 views

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 ...
10
votes
3answers
1k views

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 ...
10
votes
1answer
330 views

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 ...
10
votes
2answers
412 views

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 ...
10
votes
2answers
215 views

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
10
votes
5answers
250 views

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 ...
10
votes
1answer
296 views

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 ? ...
10
votes
2answers
227 views

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 ...
10
votes
2answers
221 views

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 ...
10
votes
1answer
502 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, ...
10
votes
2answers
251 views

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} - ...
10
votes
1answer
162 views

For each $n$, does there exist a prime $p$ and integer $k$ such that $p^k - 1$ has exactly $n$ prime divisors?

Let $n \geq 1$ be some integer. Can we always find a prime power $p^k$ such that $p^k - 1$ has exactly $n$ distinct prime divisors? For example: $n = 1$ example: $2^2 - 1 = 3$ $n = 2$ example: $5^2 ...
10
votes
1answer
112 views

Possible values of difference of 2 primes

Is it true that for any even number $2k$, there exists primes $p, q$ such that $p-q = 2k$? Polignac's conjecture talks about having infinitely many consecutive primes whose difference is $2k$. This ...
10
votes
1answer
389 views

Polar Density of a Set of Primes

In Chapter 7 of Marcus' Number Fields, he defines the polar density of a set $A$ of primes of a number field $K$ as follows: Definition: If some $n$th power of the function $$\zeta_{K,A}(s) = ...
10
votes
2answers
129 views

What are Green's almost primes?

In a general-audience talk, Ben Green explains his famous proof with Terence Tao as an application of Szemerédi's theorem, but placing the primes within a smaller set of almost-primes in which they ...