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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Limit of division vs Limit of subtraction

I was studying the Prime Number Theorem, which says $\lim_{x\to \infty} \frac{\prod(x)}{\frac{x}{\ln x}} = 1$, where $\prod(x) =$ number of primes $\leq x$. But the Wikipedia results for $\prod(x) - ...
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prime numbers - need a help

Helow, There is a question about prime numbers. Supposed that I already answer the first section. I try to answer the second section, but if n $\neq$ $2^{k}$ (for some k from the natural numbers, ...
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Relating prime numbers with irreducible polynomials using asymptotic density: is this a known theorem?

Let $p_m$ be the $m$th positive prime number in $\Bbb{Z}$. Then $f \in \Bbb{Z}[X]$ is irreducible if: $$ \liminf\limits_{m \to \infty} \dfrac{\# \{f(n) \text{ is prime } : n \lt p_m \}}{m} \gt 0 $$ ...
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1answer
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Arithmetic Progressions with a Finite Number of Primes

Is there an arithmetic progression that includes {1} that also includes only a finite number of prime numbers? Or will all progressions including {1} have infinite primes?
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Determine the quadratic character of 293 mod 379…

Determine the quadratic character of 293 mod 379. Did several other problems like this with 3, 5, 60, -1 and 307 all mod 379 but still having a tough time with this problem. I can post up work from ...
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280 views

Is this proof of the twin prime conjecture? [closed]

Identifying twin primes [1] Any natural number $n : 1<n\leq p_x^2 $ where $n$ is not divisible by any prime number less than $p_x$ is a prime number, except when $n$ is one of those prime ...
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1answer
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Can someone see the proof for that(number theory)

Please this is very important to me I would be so happy if someone is able to help... :) Let $I$ be a squarefree, natural and even number and $F$ the product of all primes $q$ where $(q-1) \mid I$. ...
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1answer
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Combinations of four consecutive primes in the form $10n+1,10n+3,10n+7,10n+9$

Here $n$ is some natural number. For example, among the primes $< 1000$ I found four such combinations: \begin{array}( 11 & 13 & 17 & 19 \\ 101 & 103 & 107 & 109 \\ 191 ...
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1answer
637 views

One of any consecutive integers is coprime to the rest

After reading this question, I conjectured a generalization of it. Conjecture: Fix $k\in \mathbb N$. Then, for all $n\in \mathbb N$, one of $n+1,\ldots,n+k$ is coprime to the rest. I ...
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Which values of $n$ is this inequality related to prime numbers true for?

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < Ap_n\prod_{i=3}^n \left(\frac{p_i-2}{p_i}\right)$$ $p$ are prime numbers and the notation $p_i$ indicates the ...
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1answer
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What's the sense behind that lemma?

Please if someone can help and can take 3 minutes I would be so so unbelievably happy because it is really important to me... Thank you :) We assume we have a $m$-th root of unity ...
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1answer
419 views

How to solve difficult positive integers and co-prime word problem?

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with derivative of algebra and prime numbers, which yields the shortest, ...
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231 views

Twin Primes (continued research)

This has become increasingly crowded, so at the onset, let me state this: My question is, is there some reason this is so linear that I'm not seeing? The only thing it seems to indicate to me is that ...
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1answer
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Determine the quadratic character of the given numbers modulo the prime 379.

a.) -1 b.) 307 I solved the same problem for 3, 5 and 60 but am having a tough time with these remaining two. Help with either one is greatly appreciated.
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2answers
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Geometric reasons finite fields have prime power orders?

All variations of proofs that finite fields have prime power orders have a very algebraic feel to them. I was wondering - is there a more geometric way to see why this is true?
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A necklace problem (related to the modulo function)

Suppose we have (non-closed) necklace with a large amount of white beads on it, and we wish to colour those beads to make a nice pattern. Unfortunately we are kind of picky about the colours we will ...
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Why are mathematicians so interested in finding out the gaps between primes and the distribution (randomness) in primes?

I'm a high school student, and I came across an article that mentioned Kanan Soundararajan's and his student's work regarding the patterns in 'random' primes. And I also read about Yitang Zhang's and ...
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Which prime numbers is this inequality true for?

Inequality What values of $n$ satisfy the following inequality? $$2(n-2) < p_n\prod_{i=3}^n \left(\frac{p_i-1}{p_i}\right)$$ Where $p$ are prime numbers and the notation $p_i$ indicates the ...
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Proof concerning Mersenne primes

Is this proof acceptable ? Lemma Let $M_p=2^p-1$ with $p$ prime and $p>2$ , thus If $M_p$ is prime then $3^{\frac{M_p-1}{2}} \equiv -1 \pmod {M_p}$ Proof Let $M_p$ be a prime , then according ...
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Proving there exists such a polynomial

I'm having trouble proving the following statement: For all primes $p$, there exists a non-constant polynomial $f(x)\in \mathbb Z_p[x]$ such that f(x) does not have a root in $\mathbb Z_p$ What I ...
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1answer
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What are the special characteristics of a natural number X where any integral number Y divided by X yields a repeating number? [duplicate]

To start off, my apologies if this question may come off as illogical. Let us take a natural number X and integral number Y and ...
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The lowest number that is a multiple of both 60 and the integer n is 180. Find the smallest possible value of n.

I have one solution but I think it's just a wild guessed one. Tell me if I am correct and also if not, then how should it be done? What I have done is divided 180 by 60 to get 3. Then take lcm of 60 ...
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Proving claim regargind upper bound of primorial

Let $\pi(m,n)$ be the number of primes in the interval $[m,n]$. Show that $\displaystyle \prod\limits_{p\in \pi(m+1,2m)}p\le {2m\choose m}$. Use the previous item to show that ...
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How do Ι show if $a^n-1$ is prime for $n >1$ then $a=2$? [duplicate]

It is well known that $a^n-1$ is prime Mersann formula for some pimes $p$ , I would like to show this implication if $a^n-1$ is prime for $n >1$ with $a$ is a positive integer then $a=2$ ? ...
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58 views

Finding integers satisfying an equation.

Suppose $p$ is a prime greater than $3$. Find all pairs of integers $(a,b)$ satisfying the equation $$a^2+3ab+2p(a+b)+p^2=0$$ A good way to start (probably) was to complete the square, ...
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Show that Z in the arithmetic progression is not compact

The Dirichlet Prime Number Theorem indicates that if $a$ and $b$ are relatively prime, then the arithmetic progression $A_{a,b}=\{...,a−2b,a−b,a,a+b,a+2b,...\}$ contains infinitely many prime numbers. ...
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What is the most efficient algorithm for factorisation when an approximate value of one factor is known

If I am given the following number: 1522605027922533360535618378132637429718068114961380688657908494580122963258952897654000350 692006139 And am told that one of ...
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226 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 ...
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1answer
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
344 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 ...
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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 ...
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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 ...
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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 ...
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1answer
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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 | ...
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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 ...
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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 ...
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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 $3x+1$ ...
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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 ...
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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 ...
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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 ...