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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Inequality with prime numbers

I found exercise in my book for number theory that I can't resolve. How do you show that $$p_n < e^{1+n}$$ where $p_n$ is $n$-th prime number?
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system of congruence - my approach

We have: $$k^3 + l^3 \equiv 0 \pmod{17}\\ k^2 + l^2 \equiv 0 \pmod{17} $$ And I get: $$k = 17n+r_k\\ l = 17m+r_l$$ And I analyzed possible rests respect to system of congruences. My result is: $$ ...
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Difference between two (not consecutive) primes

I am searching for an lower bound on the difference between the $(n+k)$-th and $n$-th prime number in terms of $k$. I have something like this in mind (conjecture): Let $(p_k)_k$ denote the ...
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Why does Euclid write “Prime numbers are more than any assigned multitude of prime numbers.”

In Euclid's Elements Book XI proposition 20 (http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html), Euclid proves that: Prime numbers are more than any assigned multitude of prime ...
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Between $n$ and $2n$ there is always a prime number. [duplicate]

Between $n$ and $2n$ there is always a prime number. I was thinking of this and looked it up on the google to find that this is true. Now, I am wondering what is the proof for it? Does any ...
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Prime larger than a twin prime

Wondered whether the following equation holds true for all twin primes such that where $a$ and $b$ are twin primes and where $b=a+2$, then $3\left[\left(\frac{a+b}{2}\right)^2-1\right]+2 = NP$. Where ...
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1answer
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Firoozbakht's conjecture and maximal gaps

In the Wikipedia article, it seems to me as if it's implied that it is enough to check the conjecture only for maximal gaps (numbers $n$ s.t. $\forall k<n:g_n>g_k$). I.e it holds that ...
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I want to get a formula for solving this

If $abc = n$ where $a,b,c,n\in \mathbb{N}$ then can you derive a formula to find the total number of triples of a,b,c as such? eg : $abc = 12$ has $4$ such triples, ...
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Factors of integers of the form $2^n-1$

I came across a problem where i had to tell the number of divisors of $2^i-1$ which are of the form $2^j-1$. I saw many contestants using the fact that if $i$ is divisible by $j$ then $2^i-1$ is ...
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$\left| (4 \mathbb{N} -1) \cap \mathbb{P} \right| \ = \ \infty$ where $\mathbb{P}$ is the set of prime numbers. [duplicate]

I try to show that there are infinitely many prime numbers in the set $ \{ 4n-1 \ : \ n \in \mathbb{N} \}$. I've been told that I needed to adjust Euclid's proof a bit so that it would work for ...
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prove if a|b and b|a then $a = \pm b$

Fairly basic I guess. Attempt: $a\neq\pm b \Rightarrow a\nmid b \vee b \nmid a$ let $a = \pm b + d, d\in \mathrm{Z} \wedge d\neq 0$ then $a\mid b \Rightarrow b\nmid a$ and $b\mid a \Rightarrow ...
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Legendre's formula

Legendre's formula counts the number of positive integers less than or equal to a number $x$ which are not divisible by any of the first $a$ primes: $$\begin{align} &\phi(x,a)=\lfloor x ...
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Does the string of prime numbers contain all natural numbers?

Does the string of prime numbers $$2357111317\ldots$$ contain every natural number as its sub-string?
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Prove that if $p$ is prime, and $a^2=b^3$

I have an exercise that I don't know how to solve. I tried to solve it in many ways, but I didn't get any progress in proving or disproving this... The exercise is: Prove or disprove: if $p$ is a ...
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1answer
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Reducing fractions with prime number denominators into additions of unities.

So I'm working on practicing reducing fractions into additions of unities (like ancient greek math). It's actually very enjoyable, except when I end up running into a fraction with a prime number as ...
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Relationship between Primes and Fibonacci Sequence

I recently stumbled across an unexpected relationship between the prime numbers and the Fibonacci sequence. We know a lot about Fibonacci numbers but relatively little about primes, so this connection ...
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Finite sequences of prime numbers

There is a lot of prime sequences: prime numbers in a special form. For example Mersenne primes are primes of the the form $2^n-1$, or Pythagorean prime are primes of the form $4n+1$. Even primes are ...
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Primorial mod $2^{32}$

Is $p_n\#$ (primorial - product of $n$ primes) periodic $\pmod{2^{32}}$? It's periodic $\pmod2$ and $\pmod4$, however it don't seems periodic $\pmod8$ and greater modulus.
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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. ...
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The number of combinations $(a,b) \in \mathbb{Z}_n \times \mathbb{Z}_n$ such that $a \cdot b = 0$

This question is about a ring for some chosen $n \in \mathbb{N}$ I wanted to find the number $M_n$ of combinations $(a,b) \in \mathbb{Z}_n \times \mathbb{Z}_n$ can be found such that $a \cdot b = 0$ ...
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series for $n$-th prime number and prime counting function

"Theoretical Computer Science Cheat Sheet" gives the following: $$p_n = n \ln n + n \ln \ln n - n + n \frac{\ln \ln n}{\ln n} + \mathcal{O}\left( \frac{n}{\ln n}\right)$$ $$\pi (n) = \frac{n}{\ln n} + ...
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Are there infinite many primes p such that 2p-1 is also prime?

I did a search online and found a similar notion called Sophie Germain prime, which by definition is a prime $p$ such that $2p+1$ is also prime. Sophie Germain primes are conjectured to be of infinite ...
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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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Consecutive Prime Problem

Consecutive primes whose quotient of their product and sum is itself a prime number. $$ 2 \times 3 \times 5 = 30 $$ $$ 30/10 = 3 $$ $$ 3 \times 5 \times 7 = 105 $$ $$ 105/15 = 7 $$ Question: ...
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Prove that there are infinitely many primes $P_i\equiv1\pmod6$

Proving that there are infinitely many primes is fairly simple: Assume that there is a finite number of primes. Let $G$ be the set of all primes $P_1,P_2,\ldots,P_n$. Compute $K = P_1 \times P_2 ...
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The perimeter of triangle $ABC$ where $|BC|=293$, $|AB|$ is a square, $|AC|$ is a power of $2$, and $|AC|=2|AB|$

In triangle $ABC$ length of side $BC$ is $293$ (a prime). If length of side $AB$ is a perfect square, length of side $AC$ power of 2 and $AC$ twice length of $AB$, find the perimeter. Kind of ...
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Is it possible to sum the divergent series with prime coefficients?

This is a follow-up of this question. It is known that the divergent series $$ P := \sum_{n=1}^\infty p_n \qquad \text{where } p_n \text{ is the $n$th prime} $$ cannot be summed by means of (prime) ...
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What are major algebraic number theory attempts, results and progressions toward Goldbach's Conjecture?

To my understanding, most progress toward Goldbach's Conjecture has been made in analytic number theory. Progress has often based on sieve, asymptotic estimation or other analytic methods. What are ...
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Can the natural numbers be defined in terms of the non-trivial zeta zeros?

Can the natural numbers be defined in terms of the non-trivial zeta zeros? Presumably they can, since $\pi(x)=\operatorname{R}(x)-\sum_{\rho}\operatorname{R}(x^\rho),$ and $\zeta(s)=\sum ...
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If an integer $a$ is coprime with an integer $b$, then will the integer $a$ also be coprime with the integer $9a + b$?

If an integer $a$ is coprime with an integer $b$, then will the integer $a$ also be coprime with the integer $9a + b$ ?
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Prime Number Primality Testing

I've developed a code and method that works in tandem with an earlier prime number algorithm I developed here: ...
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Heuristic for Dirichlet's Theorem on Arithemtic Progression

If we let $\pi_{a,d}(x) = \{p \leq x: p \mbox{ prime, } p \equiv a \mod{d}\}$ then it is a well known result that if $(a,d)=1$ then $$\lim_{x \to \infty} \frac{\pi_{a,d}(x)}{\pi(x)} = ...
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Is the Legendre sieve explicit?

The Wikipedia page for the Legendre sieve... http://en.wikipedia.org/wiki/Legendre_sieve ...says that the Legendre sieve gives upper and lower bounds on the number of primes in a given range. In ...
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Is sum of square of primes a square of prime?

I would like to know if it has been proved that : There are no $a$, $b$ and $c$, all prime numbers, such that $a^2 + b^2 = c^2$ There are no $a$, $b$, $c$ and $d$, all prime numbers, such that $a^2 ...
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The Prime Numbers Set is infinite. Is this proof correct?

Proposition: The Prime Numbers Set is infinite. Proof: Suppose we have a finite set of prime numbers $p_{1}, p_{2}, ..., p_{n}$ such that $p_{n}$ is the largest of them. Define $ c := ...
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Given all the multiples of a prime number $p \in \mathbb{Z}$, is $p\mathbb{Z}$ an ideal of $\mathbb{Z}$?

So I'm having a little trouble understanding the concept of an ideal. The book gives the "classic example" of $2\mathbb{Z}$, the even integers, saying these form an ideal. Would I be correct in ...
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Is it possible to represent subsets of natural numbers as groups with prime generators?

I'm learning group theory and I'm trying to consider the "symmetry" of a certain group of natural numbers: Here's the idea, all natural numbers are comprised of multiples of primes. So a subset would ...
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A number is a perfect square if and only if it has odd number of positive divisors

I believe I have the solution to this problem but post it anyway to get feedback and alternate solutions/angles for it. For all $n \in \mathrm {Z_+}$ prove $n$ is a perfect square if and only if $n$ ...
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Explanation of a proof of the existence of reclusive primes

The goal is to prove: For any given number $N$, there exists a prime number that is at least $N$ greater than the previous prime number and at least $N$ smaller than the following one. We call those ...
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Integers Free of Small Prime Factors

I am trying to understand (a version of) the elementary proof of the Prime Number Theorem. I've been following Tenenbaum and Mendès France's book The Prime Numbers and Their Distributions. My goal is ...
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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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prime division problem

$a,b,c \in$ {0,1,2,...,9} with at least one of $a,b,c$ nonzero. Prove that the six-digit integer $abcabc$ is divisible by at least 3 distinct primes. My thinking is not to use induction as there is ...
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Automorphisms of $\langle \mathbb{N}, \cdot \rangle$

It is an elementary fact that multiplication in $\mathbb{N}$ is commutative: $$(\forall n,m)\ n \cdot m = m \cdot n$$ This - among other things - implies that the representation of an $n \in ...
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Differences between large numbers with many factors has little factors

I apologise beforehand for the informality and lack of precision in this question but it is that way because it comes from only an intuition, nothing more than a heuristic argument. Say one has two ...
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How would you show $\pi(x)\log(1-\frac{1}{x}) \sim \frac{1}{\log x}$

How would you show $\pi(x)\log(1-\frac{1}{x}) \sim \frac{1}{\log x}$? Would you use $\lim_{x\to \infty}\frac{\pi(x)\log(1-\frac{1}{x})}{\frac{1}{\log x}} = 1$? and how would you show this? Can you ...
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Induction on prime numbers

To dive straight into the question: is there a form of induction which works on prime numbers? I've thought, and while I'm pretty sure it can be done om numbers such as even numbers or numbers ...
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A better way to prime factorize a set of numbers?

Let's say I have a range of numbers starting from 1 to 10^9 and I need to prime factorize each one of them.My basic algorithm is: 1.Use prime-sieve algorithms(Atkins or Eratosthenes(segmented ...
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$2p-2$ as the sum of consecutive prime numbers

Progress: Let $p$ be a prime such that $p≡1$ (mod 6) then $2p-2$ can be written uniquely (up to the order of addends) as the sum of some consecutive prime numbers. These are first ten examples: ...
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Do the ratios of successive primes converge to a value less than 1?

I think it's a pretty straightforward question. Does $\lim_{n \to \infty}{\frac{p_n}{p_{n+1}}} < 1?$ ***$p_n$ denotes the nth prime. Since the average gap increases between successive primes by ...
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Algorithm to identify complex Mersenne primes?

I am thinking on the complex analogue of the Mersenne primes. I think, some like a "complex Mersenne prime" could be a complex prime in the form $$2^{a+b\frac{pi}{2}i}-1$$ Where $a+bi$ is a complex ...