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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Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes.

Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes. Is there a general proof method to prove this ...
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Finding all the values of n, such that $ \varphi (n) = 12 $ [duplicate]

I have not broken this down very far. I have come to the conclusion that there are infinitely many values for n where there exists 12 coprimes to n. Since there are infinitely many primes, and primes ...
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Fundamental Theorem of Arithmetic (Canonical) missing crucial step

I've worked long on the proof of the fundamental theorem of Arithmetic and there is only one tiny piece left I can't wrap my head around. Suppose that $$\prod_{i=1}^r p_i^{m_i} = \prod_{j=1}^s ...
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Uses of Mersenne primes in math

There is an international search for Mersenne primes. The project is huge. But what are the uses of Mersenne Primes in math? Do they have any other properties other than being of the form $2^n-1$?
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Is this number composite or prime: $2000^{2002} + 2000^{2000} + 1$?

Is this number composite or prime? $$2000^{2002} + 2000^{2000} + 1$$ I want to find an easy approach to this problem.
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Normalizer and centralizer are equivalent when $p$ is the smallest prime dividing $|G|$

Let $p$ be the smallest prime dividing $|G|$, and suppose that some $P \in \mathsf{Syl}_p(G)$ is cyclic. Prove that $N_G(P) = C_G(P)$. So I let $G=p^\alpha m$ $p$ does not divide $m$. P is cyclic, ...
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Remarks on a Previous Post

Recently I have been reading this post and I have noted something significant in the fake argument. As one can easily see that the basic idea behind the argument had been to show that the sequence ...
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38 views

Is it an open problem about Riemman Hypothesis non-trivial zero? [duplicate]

Let's assume RH was correct, and $1/2+Ki$ is any one of non-trivial zero of $\zeta$, is following problem open? 1) $K$ is irrational number 2) $K$ is transcendental number
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can Sophie Germain prime be arbitrarily many?

We know that there exists arbitrarily long prime arithmetic progressions by BEN-TAO. Together with Dirichlet's theorem on arithmetic progressions, can we address that Sophie Germain prime number be ...
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Alternating series involving sums of k-primes

As an exercise, if $p_k$ are positive integers composed of k primes including repetition and $\pi_k(n)$ the number of $p_k$ not exceeding n can we show that for the alternating series of sums of ...
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Proof of $p_n<n^2$ by Elementary Means

Is there any proof of the inequality $p_n<n^2$ (for all sufficiently large $n$) by elementary means and without using Prime Number Theorem? I searched in google but in vain. The results that I ...
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Eulers totient function divided by $n$, counting numbers in the set [1,m] that are coprime to n

If we divide Euler's totient function $\omega(n)$ by $n$, we obtain a fraction. If we multiply this fraction by any natural number $m$ which gives us another natural number $p$, is it true that $p$ is ...
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Show that if $p$ is an odd prime, show a power $p^k$ can never be a perfect number

Show that if $p$ is an odd prime, show a power $p^k$ can never be a perfect number. I am little confused about this problem, any insight?
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If $m$ and $n$ are integers with $\gcd(m,n) = 1$, prove that $\sigma(mn)= \sigma(m)\sigma(n)$.

If $m$ and $n$ are integers with $\gcd(m,n) = 1$, prove that $\sigma(mn)= \sigma(m)\sigma(n)$. I am thinking about using the formula for $\sigma(p^k)$ where $p$ is prime. It follows from the ...
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Is the polynomial $X^{32} + 1$ irreducible? [on hold]

I think this question is interesting for Fermat number. Is this polynomial irreductible in $Z[X]$ ?
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On the difference between consecutive primes

Let $(p_n)$ be the sequence of prime numbers and $g_n = p_{n+1} - p_n$ Question: Is it known that $g_n \le n$? Remark: it's known that $g_n < p_n^{\theta}$ with $\theta = 0.525$ for $n$ ...
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Math for Computer Science

I have a couple of questions on the material in "Mathematics for Computer Science" by Eric Lehman and Tom Leighton. Q1. This is a theorem in the book: Theorem 24. Let $p$ be a prime. If $p|a_1a_2 ...
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About a result concerning Mersenne primes

I want to verify the proof of this result Theorem: If $p>2$ is a prime and $$H_{p}=\frac{(\sqrt3+2)^{2^{p-1}}+1}{(2^{p}-1)(\sqrt 3+2)^{2^{p-2}}}$$ is a natural number then $2^{p}-1$ is a prime ...
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Conjecture on sum of powers

Let $n$ be an odd number, $x,y$ integers and $p$ a prime number. Now, suppose that $p\ne n$ and $$ p|\frac{x^n+y^n}{x+y} $$ Then, I have been observed that $p \equiv 1 \pmod{n}$. This is, all of the ...
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Find a real $β$ such that $(β^{2^{p-1}}+1)/(β^{2^{p-2}}(2^{p}-1))$ is an integer [on hold]

Let $p$ a prime number. Find a real $β$ such that $(β^{2^{p-1}}+1)/(β^{2^{p-2}}(2^{p}-1))$ is an integer.
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Prime numbers and $\sqrt{10301}$

On my exam recently, we had the following question: Use the prime number theorem to estimate the number of primes less than $\sqrt{10301}$, and hence, give a concise argument whether 10301 is prime ...
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Riemann hypothesis: An query about the primes [on hold]

How can we describe the Riemann hypothesis easily? What is the connection between the Riemann hypothesis and prime numbers?
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Something similar to Euler's theorem

If $p$, $q$ are not equal primes. $n=pq$, $\varphi(n) = (p − 1)(q − 1)$, $d = \gcd(p − 1, q − 1)$. Is it true that for any $a$ such that $\gcd(a, n) = 1$ holds $a^{\frac{\varphi(n)}{d}} \equiv 1 ...
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Finding prime solutions to $100q+80 = p^3 + q^2$

Finding prime solutions to $100q+80 = p^3 + q^2$ Does them being prime imply some patterns on division modulo 3 or some other integer? How is this done?
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Search for very large prime (greater than $2^{57885161} − 1$) between Crystal Numbers

Denote $p[i]$ as the $i$th prime. In my opinion, the following is true: Prime Gap Axiom There are always distinct prime factors for $\{p[i],p[i]+1,p[i]+2, \dots , p[i+1]\}$. Question 1 How to ...
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Finding all natural $n$ such that $2^n+2^{2n} +2^{3n}$ has only $2$ prime factors.

Find all natural $n$ such that $2^n+2^{2n} +2^{3n}$ has only $2$ prime factors. I've tried checking the first 6-7 $n$'s on wolframalpha, but I don't see any patterns for even nor odd $n$'s. At first ...
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How prime numbers are related to special functions?

We know that the Riemann zeta function is defined as $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s},$$ for all $\Re(s)>1$. Because of Euler product formula we also know that $$\zeta(s) = ...
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Is there a way to relate prime numbers and the fourier transform

According to what I know about Fourier transforms, any continuous periodic signal can be represented as a combination of sine and cosine functions. To me, this looks analogous to the "Fundamental ...
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distribution of gaussian primes

here is a naive question that so far I don't have already found somewhere else. In the following, I consider the norm on gaussian integers with $N(a+ib)=a^2+b^2$. Consider prime gaussian integers ...
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$(n+1)^{\textrm{st}}$ prime less than $2^{2^n}$

Using elementary means, show that the $(n+1)^{\textrm{st}}$ prime is less than $2^{2^n}$ please do not use fancier stuff like the prime number theorem or beyond. using this how can you show that ...
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Discrete algebra and exponents (See body text)

Let $a,b\in\mathbb{Z}^+$. If $a \equiv b\bmod 49$, and $\gcd(a,49) = 1$. How can I find any positive integer $n > 1$, so that $b^n\equiv a\bmod 49$? I'm completely stumped by this. I've been ...
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Calculating the power of prime in factorial by changing base

The greatest power $k$ of a prime $p$ in the prime factorization of $n!$ is equal to $\frac1{p-1}(n-s(n)_p)$, where $s(n)_p$ is the sum of digits of $n$ when represented in base $p$. How to ...
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Convergence of infinite product of prime reciprocals?

Where pn is the nth prime number, does the infinite product $$\prod_{n=1}^{\infty}\left(1-\frac{1}{p_n}\right)$$ converge to a nonzero value? (Any help would be much appreciated!)
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Decryption of a RSA encrypted message is not working.

Using RSA with e=13 (encrypting power), d=17 (decrypting power) & n=33 (RSA modulus) I noticed that once I decrypted the encrypted message it would be different then the original message. Why is ...
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The representations of numbers by decimals

I'm looking for books that talk about the representation of the integers by decimals, more specifically for prime numbers. I can't found anything yet, I read something in "AN INTRODUCTION TO THE ...
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Can we find an integer $m$ such that: $2^{2p-2}-2^{p}+3=m²$

Let $p$ a prime number. Can we find an integer $m$ such that: $$2^{2p-2}-2^{p}+3=m²$$
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What makes the Mersenne primes formula more special than any of these formulas?

Mersenne Primes Formula $2^n-1$ gives false results just like any of those ones: $3^n-2, 4^n-3, P_1\cdot P_2+P_1+P_2$, or $5^n-4$ and so on.. I think that each of those formulas(including ...
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What is the number of digits of this number: $2^{333111160}$? [duplicate]

My question is: What is the number of digits of this number? : $$2^{333111160}$$
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to count the intervals

A finite set of two or more consecutive natural numbers is called a "co-prime interval" if there is no number in it that is co-prime to all other numbers in the set. Given a range [A, B], I would ...
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Can we find $n$ such that $p|2^n-1$ for a given prime $p.$

For a given prime $p$ can we find a positive integer $n$ such that $p$ is a divisor of $2^n-1.$ I know, choosing a large $n$ we can do this. But is there any proof for this? I have no idea for start a ...
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Does there exist an $a_0$ such that the sequence $a_{n+1} = 2a_n + 1$ is prime for all $n \ge 0$?

I believe I see that $a_n = 2^n(a_0+1) - 1$ but am somewhat uncertain where to proceed afterwards. I am a complete beginner at number theory and would appreciate it if someone could point me in the ...
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Question about any 2 distinct primes and the difference between their multiples

I've been thinking about the following situation. Let $p$,$q$ be two distinct primes. Let $a,b \le pq$ be any two numbers such that $a \ge b$ where $p$ divides $a$ or $b$ and $q$ divides the other. ...
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Proving that if $p$ is a prime number then $gcd (p, (p-1)!) =1$

I am just making sure whether this is a valid proof: Since $p$ is a prime number, then $p$ is only divisible by $1$ or $p$ Suppose we want to take the $gcd (p,a)$ with a, an arbitrary ...
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Is it true that $2^{p}-1$ is a prime number?

Let $p$ be an odd prime such that $$p \equiv 1 \pmod{4}$$ and $p$ and $p-2$ form a twin prime pair. My question: Is it true that $2^{p}-1$ is a prime number?
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If $p,q$ are prime, $q$ odd $p \not\equiv 1 \pmod q$, is there an integer $x$ such that $p\mid 1+x+\ldots+x^{q-1}$

Suppose $p,q$ are two distinct prime numbers, $q \geq 3$ and $p \not\equiv 1 \pmod q$. Then I have the following problem: Prove that there is no integer $x \in \mathbb{Z}$ such that ...
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About a recurrence equation of prime numbers

Let $p$ be a prime. Consider the recurrence equation $$s_{n}=(s_{n-1}²-2)(mod(2^{p}-1))$$ where $s₀=4$ My question is: Can we write this recurrence as follow? $$s_{n}=(2^{p}-1)q+(s_{n-1}²-2)$$ ...
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Can this be solve using modular arithmetic? $k$ is prime $\Rightarrow$ $8k+1$ is prime

Is the following statement true or false? $\forall k \in \mathbb{N}, k$ is prime $\Rightarrow$ $8k+1$ is prime The answer is that the statement is false because if $k=7$, then $k$ is prime but ...
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Primality of Stirling numbers of second kind

Apart from the Mersenne primes $M_p=2^p-1=S(p+1,2)$, and the four primes $S(n,4)$ where $n$ is given in http://oeis.org/A100958, are there other Stirling numbers of the second kind $S(n,k)$ which are ...
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Which of the following is true?

Let $\hspace{0.2cm}$$p,q,r$$\hspace{0.2cm}$ be prime numbers greater than 100,then which of the following is true? $3|p^2+q^2+r^2$ $q|p^5$ There exists integers $x,y$ such that ...
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Can we have $\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$ for some constant $c>0$, where $x>1.$

Let positive interger $n$ is square-free, that is $n=p_1p_2\cdots p_r$ some $r$. Can we have $$\sum_{n\leq [x]}e^{-\sqrt{\frac{\log x}{r}}}\ll \frac{x}{e^{c \sqrt{\log x}}}$$ for some constant ...