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.

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Proof for Carmichael theorem

if $n=p_1^{a_1}p_2^{a_2}p_3^{a_3}\dots p_r^{a_r}$ and $\lambda(n) = lcm[(p_1-1)(p_1^{a_1-1}),(p_2-1)(p_2^{a_2-1}),(p_3-1)(p_3^{a_3-1}),\dots,(p_r-1)(p_r^{a_r-1})]$ then $k^{\lambda{n}} \equiv 1(...
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Extrema of the Ratio of Consecutive Primes

Let $p_i$ denote the $i$th prime number. We know that $\frac{p_{n+1}}{p_n}\rightarrow 1$ as $n\rightarrow\infty$. Therefore, if we pick some real number $c>1$, there should be some positive integer ...
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The binomial coefficients $\binom{n}{ p}$ are divisible by a prime $p$ only if $n$ is a power of $p.$

I'm looking for a "high school / undergraduate" demonstration for the: All the binomial coefficients $\binom{n}{i}=\frac{n!}{i!\cdot (n-i)!}$ for all i, $0\lt i \lt n$, are divisible by a prime $p$ ...
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How many $\overline{a}\in\left(\mathbb{Z}/91\mathbb{Z}\right)^\times$ pass the Fermat and Miller-Rabin primability tests?

Let $$\text{F}_{91}:=\left\{\overline{a}\in\left(\mathbb{Z}/n\mathbb{Z}\right)^\times:91\text { passes the Fermat primality test to base }a\right\}$$ and $$\text{MF}_{91}:=\left\{\overline{a}\in\left(\...
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Twin primes : prove the convergence of $ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} (\log(p)+\frac{1}{p})²$

let $T$ be the twin primes set : $p \in T $ if and only if $ p$ and $p+2$ are primes. Can you help me establish the convergence of : $$ \lim_{N\sim\infty} \frac{1}{N} \sum^{N}_{p\in T} (\log(p)+\frac{...
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(p-1)! number theory problem

I was working on this little problem: Let $\frac{a}{(p-1)!} = 1 + 1/2 + 1/3 + ... + 1/(p-1)$, where p is a positive prime. (a) Prove $p\mid a$. (b) Can $p^2 \mid a$? I thought (a) was pretty self-...
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101 views

Combining primes for getting primes?

I was thinking about what would happen when we combine two prime numbers $p$ and $q$ into one number $:pq:$ . Like if $p=5$ and $q=3$ , then $:pq:=53$ . Then if $p=7$ and $q= 11$ then $:pq:=711$ and ...
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Is every prime the average of two other primes?

$\forall {p_1\in\mathbb{P}, p_1>3},\ \exists {p_2\in\mathbb{P},\ p_3\in\mathbb{P}};\ (p_1 \neq p_2) \land (p_1\neq p_3) \land (p_1 = \frac{p_2+p_3}{2})$ Now I'm not a 100% sure about this, but I ...
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Why all composite numbers have this property?

Define $f(n)=\sum\limits_{A \in S} f_{1}(n,A),\ n>2,\ n \in \mathbb{Z}$, where $S$ is the power set of $\{\frac{1}{2},\cdots ,\frac{1}{n-1}\}$. Define $\ f_1(n,\varnothing)=1,\ f_{1}(n,A)=(-1)^{\...
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Proof of infinitely many primes, clarification

Proof: The proof is by contradiction. Suppose there are only finitely many primes. Let the complete list be $p_1,p_2,\dots,p_n$. Let $N = p_1p_2 \dots p_n+1$. According to the Fundamental Theorem of ...
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Primes in a concave sequence

Are there infinitely many consecutive tuples of primes $(p_{n-1},p_n,p_{n+1})$ satisfying $$\frac{1}{2}(p_{n-1}+p_{n+1})\le p_n$$ I don't know how to do this problem. Maybe we use the prime number ...
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2answers
114 views

How can I solve this problem without having to do it by hand?

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
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555 views

Infinite sum of prime reciprocals

Let $\mathbb D$ be the set of all real numbers that can be expressed as a sum of distinct prime reciprocals, i.e. $\mathbb D = \{ d \in \mathbb R \mid d = \sum_{k \in \mathbb K} \frac 1k $ for some (...
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How can I solve this problem without doing it by hand? [duplicate]

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement without forcing me to do it ...
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2answers
106 views

Is there any way to solve this problem without having to do it by hand? [duplicate]

I'm dealing with the following problem in computational programming. I'm trying to find a way to build an algorithm that can quickly resolve the following problem statement. Is there any way to group ...
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52 views

$\log$ approximation for $\pi(x)$

It seems that a reasonable $\log$ approximation for $\pi(x)$ can be given, where $f(y, x) := \log\left(\dfrac{\log(x)}{\log\left(e(y - \lfloor y\rfloor) + x^{1/x}(1 - y + \lfloor y\rfloor)\right)}\...
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92 views

Finding Prime triples with $p_{n} +p_{n+1} −p_{n+2} = 1$

I was just looking at a sequence of primes and suddenly I got this thought that $p_2 +p_3 −p_4 = 1$ since $p_2 = 3, p_3 = 5, p_4 = 7$. Also for $p_3 = 5, p_4 = 7, p_5 = 11$ one has $p_3 +p_4 −p_5 = 1$....
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171 views

Intuitively, what separates Mersenne primes from Fermat primes?

A Mersenne prime is a prime of the form $2^n-1$. A Fermat prime is a prime of the form $2^n+1$. Despite the two being superficially very similar, it is conjectured that there are infinitely many ...
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On non-divising primes of an integer $x$

We know more about divisor than non-divisors, If we consider the sets : $$P^{1}_{x} =\left \{ p \leq x : \ p \in \mathbb{P} \right \}$$ ($\mathbb{P} $ is the primes set) $$ P^{2}_{x} =\left \{ p_{i}...
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61 views

Relative error in prime number fitting

I got curious about what it would look like if I made a scatter plot of the n-th prime number as a function of n (lets call it p(n), so that p(1)=2, for instance ). Not being an expert in the area I ...
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65 views

How many natural value of n such that $n^5+2n^4+n-1$ is prime number?

From above polynomial, I can only get one value to make it prime. The value, I guess, is only one. For $n=1$, we got: $$(n^5+2n^4+n-1)= 1+2+1-1= 3 \quad\text{(prime)}$$ But, I cannot find the ...
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Solution of a equation in natural number nvolving reciprocal of prime

Let $p$ be a prime and $n$ a natural number . Solve in $\mathbb{N}$ the equation $$\sum_{k=1}^{n}\frac{1}{x^k_k}=\frac{1}{p}$$
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Conjecture: If $ N $ and $ N + 10 $ are prime, then $ N + 20 $ is composite.

Conjecture: If $ N $ and $ N + 10 $ are prime, then $ N + 20 $ is composite. Here are some examples: $ 19 $ and $ 29 $ are prime; $ 39 $ is composite. $ 241 $ and $ 251 $ are prime; $ 261 $ is ...
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24 views

Showing $f\in\mathbb{F}_{p^d}[X]:f'=0\Rightarrow\exists g\in\mathbb{F}_{p^d}[X]:f=g^p$

Let $\mathbb{F}_{p^d}$ denote the final field with $p^d$ elements and $\mathbb{F}_{p^d}[X]$ denote the polynomial ring in $X$ over $\mathbb{F}_{p^d}$. How can we show $$f\in\mathbb{F}_{p^d}[X]:f'=0\...
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207 views

The most efficient method for generating new prime numbers

What is the most efficient method for generating a prime number larger than the largest known prime number, and what is the complexity of this method? Techniques considered: Mills' Constant - ...
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166 views

References for Legendre's prime-counting function

This question is about Legendre's prime-counting function, the one that can be used to calculate the exact amount of prime numbers that are less than or equal to a given number (as long as the number ...
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492 views

least common multiple $\lim\sqrt[n]{[1,2,\dotsc,n]}=e$

The least common multiple of $1,2,\dotsc,n$ is $[1,2,\dotsc,n]$, then $$\lim_{n\to\infty}\sqrt[n]{[1,2,\dotsc,n]}=e$$ we can show this by prime number theorem, but I don't know how to start I ...
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Find all $n$ so that $c_n$ $>$ $\pi(n^2)$

Find all $n$ $\in$ $\mathbb{N}$ so that $p_{c_n}$ $>$ $n^2$ where $p_n$ denotes the $n$-th prime and $c_n$, the $n$-th composite. I have tried doing the problem using The stronger version of ...
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Alternative Proof of Infinitely Many Primes? [duplicate]

I've seen Euclid's proof of infinitely many primes, what are other approaches to proving there are infinitely many primes?
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If there is an $a\in\mathbb{Z}$ with $a^{n-1}\equiv 1\mod n$ but $a^{\frac{n-1}p}\not\equiv 1$ for all primes $p\mid n-1$, then $n$ is a prime

Let $n\in\mathbb{N}$ with $n\ge 3$ and $a\in\mathbb{Z}$ such that $$a^{n-1}\equiv1\text{ mod } n\;\;\;\wedge\;\;\;a^{\frac{n-1}{p}}\not\equiv1\text{ mod }n\;\;\;\forall p\in\mathbb{P}:p\mid n-1$$ ...
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Conjecture about prime numbers

$$\forall k\in\mathbb{N},k\ge1,\exists p:k^3\lt p\lt (k+1)^3$$ with $p$ prime number. In other words is it possible to prove that for every $k\gt1$, with $k$ integer number it exists a prime number ...
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How to prove this??

Does the following inequality hold? $p(n)\leq 2^n,$ where $p(n)$ is the $n$th prime. If this is true then it follows that: If $p(n)=p(m)^x+p(o)^y$, then $\max[x,y] \le n$.
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Euler totient of a number

If $n= \prod_{i=1}^{m} p_i$, all $p_i$ pairwise distinct, then number of coprimes below $n$ is $\prod_{i=1}^{m} (p_i-1)$. For example with $m=2$, there are $p_2-1$ multiples of $p_1$ below $n$ and ...
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115 views

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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128 views

Under assumption that $\frac{M_{n+1}}{M_n} \le 2$, what is true?

This question was hinted upon with the still open question at [1]. Let $M_n = $ A005250($n$) of the OEIS. That is to say, $M_n = p_{i+1}-p_i$, where $p_i$ is the smallest prime such that $p_{i+1} - ...
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Prove that $512^3 + 675^3 + 720^3$ is a composite number

We have to prove that the number $$N=512^3 + 675^3 + 720^3$$ is composite. I tried to use the identity $(a^3+b^3+c^3)=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)+3abc$ hoping to take out some common ...
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If p is a prime positive integer, find all subfields of GF(p)

If p is a prime positive integer, find all subfields of GF(p) This question just seems too vague.
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1-to-1 correspondence between twin primes and $n^2-1$

I am trying to establish the one-to-one correspondence of twin primes to integers $n$ where $n^2-1$ has 4 divisors. It is clear to me that this is the case, since $$n^2-1=(n+1)(n-1)$$ where the RHS ...
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(Easy?) consequence of the Riemann Hypothesis

I'm trying to show that the relation $\psi(x)=x+O(\sqrt{x}\log ^2 x)$ (consequence of the Riemann hypothesis) implies $\pi(x)=Li(x)+O(\sqrt{x}\log x)$, where $Li(x)=\int_2^x \frac{dt}{\log t}$. I ...
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Relationship between the Carmichael function and Euler's totient function

Let $\lambda$ denote the Carmichael function and $\varphi$ Euler's totient function. Furthermore, let $p$ denote any prime number and $k\in\mathbb{N}$. The wikipedia article about $\lambda$ states: $$\...
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Beginner Proof about Primes

I am interested in understand the proof of infinitely many primes. It seems like quite an easy proof, ( I know there are many but I am referring to the proof that goes as follows); " Suppose there ...
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Any simpler way to do Pollard's p-1 method?

I found calculating factorization by Pollard's p-1 method is almost impossible if use a conventional scientific calculator. For example, I am trying to factor ...
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Proof that there are infinitely many primes of the form $6k+1$. Proof verification

Theorem. there are infinitely many primes of the form $6k+1$. I've just proved that there are infinitely many primes of the form $6k+1$. Could you please check my proof? At first, I proved that ...
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On Newman/Zagier's proof of PNT

I have just got this paper: http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf and I have a serious doubt: When proving that soft Tauberian theorem he explicitly uses ...
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Divisibility of sum of exponents

Consider the sequence $$r, \ ra, \ ra^2, \ ra^3, ... \ , ra^n \mod M $$ such that: $$ ra^{n+1} \equiv r \mod M$$ and $a \ne 1$ and $a,r$ are both coprime to $M$ Is it always true then that: ...
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Zhang's theorem and Polignac's conjecture

Yitang Zhang made a groundbreaking discovery when he proved that there are infinitely many pairs of prime numbers which differ by less than $70,000,000$. Zhang's theorem has been significantly ...
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59 views

Number of the form $3k+2$ has a prime factor of the same form

I was thinking of using a proof by contradiction. I will also note that following the book I am using has not covered congruences in case that is the methodology. Let $n=3k+2=ab$. Assume that $n$ ...
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A polynomial formula for the primes

Is there a proof that there is no polynomial which would return $n$th prime for the input value $n$? In other words is there an explanation for why there is no polynomial $P(x)$ such that $P(n)=p_n$ ...
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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 to ...
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Primes as a difference of powers

Find the smallest prime that cannot be written as $$|3^a - 2^b|$$ EDIT: I forgot to mention that $a$ and $b$ are whole numbers. I tried to expand $3^a$ as $(2+1)^a$ using binomial theorem but ...