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.

learn more… | top users | synonyms

3
votes
2answers
225 views

Product of its Prime Factors

Given that $4095 = 8^4 - 1$ write $4095$ as a product of its prime factors. I know how I could separate $4095$ into prime factors however I'm not sure how I could use $8^4 - 1$ to help me. I could ...
1
vote
3answers
193 views

$n$th prime number

Is there an explicit formula that tells us what the $n$th prime number is if we know what the $nth$ squareful number is? If there is, what are its applications, and if there isn't, what use would such ...
3
votes
2answers
98 views

Is the concatenation of all prime numbers, a universe/disjunctive number? [duplicate]

A universe number is a number which contain any finite length string of digits for a base. Reference here. These are usually called "disjunctive numbers" in English. Then the question is, is the ...
2
votes
4answers
12k views

What is the sum of the prime numbers up to a prime number $n$? [duplicate]

How to find the sum of prime numbers up to a prime number $n$, that is for example: the sum of prime numbers up to 7 is: 2+3+5+7=17. So what is the formula for finding: $$\sum_{k=0}^n p_k=????,$$ ...
1
vote
1answer
66 views

Proving if $2^n + n^2$ is a prime, show that $n ≡ 3 \pmod 6$

If $n$ is a positive integer greater than $1$ such that $2^n + n^2$ is a prime, show that $n ≡ 3 \pmod 6$ Source of the question : http://math.stanford.edu/~paquin/ModPS.pdf I tried this for hours ...
52
votes
16answers
16k views

For any prime $p > 3$, why is $p^2-1$ always divisible by 24?

I know this is very basic and old hat to many, but I love this question and I am interested in seeing whether there are any proofs beyond the two I already know.
1
vote
1answer
43 views

Let $p_n = n$th odd prime. When is $p_n$ a continuous function of $n$?

Under what topologies is the function $p(n) = n$th odd prime continuous? If we take the Euclidean topology on $\Bbb{R}$ and induced it onto the subspace $\Bbb{N}$ and called it $\tau$. Then isn't ...
2
votes
0answers
55 views

Erdos' Arithmetic Progression of Polignac's Numbers

Paul Erdos has proven that there is an infinite arithmetic progression of Polignac's numbers - odd numbers that cannot be represented as a sum of a prime and a power of $2$, in Erdos, Paul. "On ...
1
vote
2answers
79 views

Twin Primes, their Arithmetic Means and some properties.

These are two problems which I have been trying to solve. The arithmetic mean of twin primes 5 and 7 is 6 which is a triangular number. Do there exist any other such twin primes? If they exist ...
-1
votes
0answers
27 views

weighted graph for representing integer

I was not able to find references by googling for a while, so my question is this, does anyone had this idea? Let us consider the following graph $G_k(V,E)$ where $V=\{v_i | v_i \in \mathbb{N}, 2 ...
6
votes
1answer
284 views

What is the infinite product of (primes^2+1)/(primes^2-1)?

I have shown that the infinite product $$\prod_{p \in \mathcal{P}}\frac{p^2+1}{p^2-1}$$ is equal to $\frac{5}{2}$ (pretty remarkable!). I have checked this numerically with Wolfram Alpha for up to ...
15
votes
4answers
1k views

Prove there are no prime numbers in the sequence $a_n=10017,100117,1001117,10011117, \dots$

Define a sequence as $a_n=10017,100117,1001117,10011117$. (The $nth$ term has $n$ ones after the two zeroes.) I conjecture that there are no prime numbers in the sequence. I used wolfram to find the ...
5
votes
0answers
37 views

find a 5-digits $q$ such that $2^q+17$ be a prime [closed]

How we can find a 5-digits number $q$ such that $2^q+17$ be a prime number? Does there exist such number?
1
vote
1answer
24 views

Prime solutions to an exponential diophantine equation

Suppose that $a,n,t$ are positive integers greater than $1$ and $q$ is a prime number. I write $a_{n} = \binom {a^{n}} {a}$. In general I was curious about $a_{n}-k =q^{t}$ in particular the case ...
29
votes
1answer
343 views

A spiralling sequence based on integer divisors. Has anyone noticed this before?

Firstly, please excuse the informal style of my explanation, as I am not a mathematician, although I am aware that this can be explained in more formal terms. I have mapped integers to points on a ...
2
votes
1answer
68 views

Consecutive prime numbers multiplication pattern

Playing with primes in excel I came to a pattern that I do not understand and I would like to know more about it. Example: |Prime numbers | Multiplies | Subtraction of | Difference of | ...
4
votes
0answers
82 views

Effect of 'Prime conspiracy' on the fact that prime numbers are the generators of integers [closed]

In Unexpected biases in the distribution of consecutive primes, the authors have discovered that prime numbers have decided preferences about the final digits of the primes that immediately follow ...
1
vote
3answers
37 views

How do I prove that “If prime p does not divide natural number m, then gcd(p,m) = 1”

I am having a problem with this. If prime p does not divide natural number m, then gcd(p,m) = 1 I had to use this for my another proof and because I thought it was quite intuitive, I just assumed ...
4
votes
2answers
97 views

Unable to find solution for $a^2+b^2-ab$, given $a^2+b^2-ab$ is a prime number of form $3x+1$

I have a list of prime numbers which can be expressed in the form of $3x+1$. One such prime of form $3x+1$ satisfies the expression: $a^2+b^2-ab$. Now I am having list of prime numbers of form $3x+1$ ...
6
votes
1answer
71 views

Part of proof of Prime Number Theorem

If $x\ge 1$, let $\pi(x) = \sum_{p\le x} 1$ denote the number of primes $\le x$. The prime number theorem states that $\pi(x) \sim {x\over \log(x)}$ This is usually proved by studying the related ...
15
votes
3answers
1k views

Use this sequence to prove that there are infinitely many prime numbers. [duplicate]

Problem: By considering this sequence of numbers $$2^1 + 1,\:\: 2^2 + 1,\:\: 2^4 + 1,\:\: 2^8 +1,\:\: 2^{16} +1,\:\: 2^{32}+1,\ldots$$ prove that there are infinitely many prime ...
1
vote
3answers
57 views

Prove that every rational solution of $x^n=c$ is integer

Prove that every rational solution of $x^n=c$ is integer $c,n\in \mathbb N$ My start: Let $x=u/q\quad \text{ such that } \gcd(u,q)=1$ $$\left( \frac u q \right)^n=c $$ I realy don't ...
0
votes
1answer
48 views

If $b^3\mid c^2\qquad c,b\in \mathbb N$ prove that $b\mid c$

If $b^3\mid c^2\qquad c,b\in \mathbb N$ prove that $b\mid c$ What I did: $b=p_1^{\alpha 1}\cdot p_2^{\alpha 2}\cdot \dots \cdot p_k^{\alpha k}$ $c=p_i^{\beta 1}\cdot p_{i+1}^{\beta 2}\cdot ...
20
votes
2answers
344 views

Is there a polynomial such that $F(p)$ is always divisible by a prime greater than $p$?

Is there an integer-valued polynomial $F$ such that for all prime $p$, $F(p)$ is divisible by a prime greater than $p$? For example, $n^2+1$ doesn't work, since $7^2+1 = 2 \cdot 5^2$. I can see that ...
2
votes
1answer
97 views

Largest prime gap under $2^{64}$

Thanks to Tomás Oliveira e Silva's extensive calculations, it is known that the largest prime gap less than $4\cdot10^{18}\approx2^{61.8}$ is 1476. I'd like an upper bound for the largest prime gap ...
0
votes
0answers
50 views

Whose theorem is this?

Theorem. If $x$ is composite, it has two factors $k,a$ such that $x=ka$ and $x\gt k\ge a$. If $k$ is the greatest factor of $x$, $a$ is prime. Proof. $k$ is the greatest factor of $x$, so $a$ is ...
1
vote
0answers
29 views

Why does $\equiv 1\ (\text{mod}\ n)$ seem so important?

I'm not great with math so please feel free to correct any mistakes in my question (or add more examples). I'm a software engineer and have recently wanted to better understand the maths behind RSA ...
6
votes
1answer
109 views

Proportion of elements of prime order $p$ in $S_n$

I was trying to answer the following question recently : What is the proportion of elements of order $p$ in the symmetric group $S_n$ , where $p$ is some prime number ? I managed to work out that in ...
1
vote
1answer
47 views

What's about $\sum_{k=1}^{n-1} p_{k} \sum_{l=k+1}^{n} p_{l}$ for prime numbers?

By specialization of this formula, here in PROBLEMA 36, page 453 (in spanish), taking $\frac{1}{x_i}$ as the ith prime number we've (with at least two summands) $$ \left( \sum_{k=1}^{n} p_{k} ...
2
votes
1answer
35 views

Bound for Chebyshev function

Consider the Chebyshev function defined as: $\psi(x)=\sum\limits_{n\leq x} \Lambda(n)$, where $\Lambda(n)=\log p$ if $n$ is a power of some prime $p$ and is equal to $0$ otherwise. Could someone ...
3
votes
2answers
180 views

Is $1000000000000066600000000000001$ (Belphegor's prime) actually a prime?

There is a Wikipedia article about that evil Belphegor's prime, but the references there seem relatively weak. Is this number actually a prime?
1
vote
3answers
159 views

How to prove the number is a prime?

A natural number $n$ has the property that if $d$ divides $n$ then $d+1$ divides $n+1$. Show that $n$ must be a prime.
34
votes
6answers
23k views

Is there a known mathematical equation to find the nth prime?

I've solved for it making a computer program, but was wondering there was a mathematical equation that you could use to solve for the nth prime?
1
vote
0answers
32 views

Constructing a smooth function whose roots consist only of each of the primes.

My first attempt: $$f(x) = \prod_{i=1}^\infty \left(1- \frac x {p_i} \right)$$ If we take a look at the Riemann zeta function: $$ \zeta(s) = \sum_{n = 1}^\infty \frac 1 {n^s} = \prod_{i = 0}^\infty ...
8
votes
3answers
18k views

How to find number of prime numbers between two integers

I have two integers, $x$ and $y$ so that $x \lt y$. How many prime numbers are there between $x$ and $y$ (exclusive). Is there a formula or algorithm to compute?
1
vote
0answers
23 views

Let $m =\prod_{i=1}^{r} p_i^{α_i}$, with $α_i \ge 1$ and $p_i \ge 3$ for each $i$, be the canonical representation of $m$ and…

Let $m =\prod_{i=1}^{r} p_i^{α_i},$ with $α_i \ge 1$ and $p_i \ge 3$ for each $i$, be the canonical representation of $m$ and let $a$ be relatively prime to $m$. Show that $x^2 \equiv a \pmod m$ is ...
1
vote
2answers
45 views

Notation without cases? $f(x)=\begin{cases}p,&\text{if $x=p^k$}\\1,&\text{otherwise}\end{cases}$

Is there any other way to write the function $f:\Bbb N\to\Bbb N$ such that $$f(x)=\begin{cases}p,&\text{if $x=p^k$}\\1,&\text{otherwise}\end{cases}$$ when $p$ is prime and $k\in\Bbb N$?
0
votes
2answers
75 views

Polynomials generating primes.

Does there exist a polynomial $f(x)$ with integer coefficients with $f(0)=1 or -1$ such that $f(x)$ is a prime for all $x$ in $N$? If $x$ can be $0$ ,then, it is easy. But I cannot get it if $x$ ...
1
vote
0answers
34 views

Primes congruent to {0,2} modulo 3 and the greatest common divisor of $2s(s-1)$

This problem is related to the question asked here: The primes $s$ of the form $6m+1$ and the greatest common divisor of $2s(s-1)$ I would like to prove the following. If $s$ is prime Let $\psi(s) = ...
2
votes
4answers
126 views

For every positive integer $n$, $n^2 + n +19$ is prime

I'm trying to prove that for every positive integer $n, n^2 + n +19$ is prime. I tried to disapprove it saying that is is not prime. If it's not prime, then $n^2 + n +19$ has to have at least two ...
1
vote
2answers
25 views

Can the product of two Fermat witnesses where the witnesses are prime be a Fermat liar?

Let $n$ be composite. If $a$ is coprime to $n$ such that $a^{n-1} \equiv 1 \pmod n$ then $a$ is called a Fermat liar. If $a^{n-1} \not\equiv 1 \pmod n$, then $a$ is a Fermat witness to the ...
4
votes
1answer
440 views
1
vote
0answers
22 views

On a theorem of Hensel

In the paper Binomial coefficients modulo prime powers, Andrew Granville state the following theorem: Let $n, m$ and $r=n-m$ be three given positive integer and $p^k$ is the exact power of $p$ ...
-2
votes
1answer
45 views

Let $a_1 = 2$ and for all natural number n, define $a_{n+1}= a_{n}(a_{n}+1)$. Then as $n\rightarrow \infty$, the number of prime factors of $a_{n}$ [closed]

Let $a_1 = 2$ and for all natural number n, define $a_{n+1}= a_{n}(a_{n}+1)$. Then as $n\rightarrow \infty$, the number of prime factors of $a_{n}$: goes to infinity. goes to a finite limit. ...
1
vote
0answers
46 views

Sum of first n primes [duplicate]

Can we claim it is asymptotic to $n^2\log n$? I argue that because $p_n\sim n\log n$, we can say: $$\sum_n n\log n=\log1+2\log2+\dots+n\log n$$ $$=\log1+\log2+\dots+\log n$$ $$+\log2+\dots+\log n$$ ...
4
votes
0answers
21 views

What is known about the counting function of Gaussian primes"

The counting function of primes among $\Bbb{N}$, describing the asymptotic density of the primes, is well known (the Prime Number theorem). Let's define a mild generalization of the counting function ...
0
votes
1answer
31 views

What are the solutions to this equation (primes, modular arithmetic)?

Given: $m,n\in\mathbb{N}$ and $p$ is prime. Find the solutions to the following equation: $$m^2-3mn+(np)^2=12p$$ Thank you in advance.
1
vote
1answer
87 views

Possible divisors of $s(2s+1)$ follow up question.

This question is related to this post:Possible divisors of $s(2s+1)$. I have some follow up questions which should be a new post. I write $\psi(s) = s(2s+1)$. We showed that for every prime $s$ that ...
1
vote
0answers
35 views

Distribution of final digits of consecutive primes

There's been a lot in the press recently about the unexpected distribution of final digits in pairs of consecutive primes, and many people have written programs to confirm the observation that pairs ...
9
votes
1answer
425 views

Improving Zhang's prime gap

I am referring to Zhang's paper. Since the set $\cal{H}$ is a subset of $[3.5\times 10^6, 7\times 10^7]$, shouldn't the prime gap he obtained be less than $ 7\times 10^7 - 3.5\times 10^6$ rather than ...