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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Concerning types of square-free numbers and comparing sizes of their subsets.

Call a square-free a 2-prime if it has exactly two prime divisors. Call a square-free a 3-prime if it has exactly three prime divisors,etc. Does there exist an integer sufficiently large N such that ...
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Concerning types of square-free numbers.

Call a square-free number a 3-prime if it is the product of three primes. Similarly for 2-primes, 4-primes , 5-primes, etc. Are there two consecutive 3-primes with no 2-prime between them?Are there ...
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Numerical verification of the ternary Goldbach conjecture

In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this ...
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Why there are no other known Fermat primes.

Fermat primes are prime numbers of the form $2^{2^n} + 1$: $$3,~5,~17,~257,~65537$$ There are no other known Fermat primes. But why?
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elementary proof that infinite primes quadratic residue modulo $p$

$p \gt 2$ is a prime, then there are infinite primes $q$ such that $q$ is a quadratic residue modulo $p$. With Dirichlet's theorem on arithmetic progressions, the problem is easy! How about ...
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Primality of the number of obtained by concatenating the n consecutive digits

Let $f_n$ be the number obtained by concatenating the first $n$ numbers (in base 10). For example $f_1 = 1, f_3 = 123$ and $f_{13} = 12345678910111213.$ Now if $n$ is even or divisible by $5$ then ...
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Is it true that $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$?

Let $p$ be a prime number, are the following statements true? 1.Quadratics of the form $6n^2+p$ gives primes for $n=0,1,2,\dots,p-1$ iff $Q(\sqrt{-6p})$ has class number $4$. And all such primes ...
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Creating Polynomial

By relative prime factor theorem $$R = (Zm,+,.)$$ where R is the ring structure the input is $e_0 = 0$ and $e_1=1$ output is $$S_0 = { k : \gcd(m,k)>1 }$$ $$S_1 = { k : \gcd(m,k) = 1}$$ Now ...
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Factor factorials

How would you find the greatest prime factor of a factorial? For instance, 82! The 2 and 41 that are yielded when you prime-factor 82 seem to have no correlation to the prime factorization of 82!
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Largest prime factor

Let $$ n = (1^2 - 0^2) * (2^2 - 1^2) * (3^2 - 2^2) * (4^2 - 3^2) * ... (100^2 - 99^2).$$ What is the largest prime that divides n? Please explain how to go about solving this, for I have never seen ...
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Let $u_{n+3} = u_n + 2u_{n+1}$ . Show that $p$ divides $u_p$ for all $p$ prime number.

Let $(u_n)$ a sequence such that $u_0 = 3$, $u_1 = 0$, $u_2 = 4$ and $u_{n+3} = u_n + 2u_{n+1}$ Show that $p$ divides $u_p$ for all $p$ prime number. I'm really stuck on this exercise, Does ...
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Siamese Twin primes

Can someone edit my answer to this question whether I am answering the question or I am not? The question is Let us say that two prime numbers $p$ and $q$ are siamese twins if $|p-q|=1$. List all ...
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Confused by a step in a proof that $a^x - b^y = c$ has at most two solutions in positive integers $x,y$

The theorem is Theorem 1.1 from Michael A. Bennett in his "On Some Exponential Equations of S.S. Pillai". Here is the statement of the theorem: Theorem 1.1. If $a,b,c$ are nonzero integers with $a,b ...
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Arithmetic progression of primes question

Is it known whether for all positive integers $k$ there is an integer $a$ such that $a+30n$ is a prime number for all $n\in \{1,\ldots,k\}$?
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Riemann's zeta as a continued fraction over prime numbers.

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers. $$ \zeta(s)=1 +\cfrac{\frac{1}{2^{s}}}{...
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Difference between sum of first n primes and prime(prime(n))

The seq is: -1, 0, -1, 0, -3, 0, -1, 10, 17, 20, 33, 40, 59, 90, 117, 140, 163, 218, 237, ... http://oeis.org/A239731 Is there's a formula looks like $$a(n) =n^2logn/2$$ for this seq?
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In $1 < k < n-10^6$, what is $k$? (details in question)

This is a homework question of mine, I am not searching for the solution but rather what it means. It seems pretty straight forward but I am a little confused as to what the $k$ in $1 < k < n-10^...
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How does one compute how big the cycle of modding by a prime number is?

If I take the $k \in \mathbb{N}$ power of 10 and mod it by a large prime, I notice that the remainders repeat at some point. For instance $10^k mod~7$ seems to repeat every $8$th value of $k$. Given ...
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Number of primes in $[30! + 2, 30! + 30]$

How to find number of primes numbers $\pi(x)$ in $[30! + 2$ , $30! + 30]$, where $n!$ is defined as: $$n!= n(n-1)(n-2)\cdots3\times2\times1$$ Using Fermat's Theorem: $130=1\mod31$, (since $31 \in \...
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First 10-digit prime in consecutive digits of e

Problem. What is the first $10$-digit prime in consecutive digits of $e$. For those of you who don't know, in 2004 the answer produced a URL to a Google employment page (sort of). I just found about ...
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Need help in understanding $ord_p{a}$ as used in Theorem 1.1 from “On Some Exponential Equations Of S. S. Pillai”

I have a question about very early argument in the proof of Thereom 1.1. Theorem 1.1 of On Some Exponential Equations of S.S. Pillai states that if $a,b,c$ are nonzero integers with $a,b \ge 2$, then ...
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Prime numbers like 113

The number 113 is prime. The sum, product and all permutations of it's digits are prime. Are there any other such prime numbers?
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Truncatable primes

Why only 11 ? The number 3797 has an interesting property. Being prime itself, it is possible to continuously remove digits from left to right, and remain prime at each stage: 3797, 797, 97, and 7. ...
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Solve for $p^a + 1 = 2\cdot q^b$ where $p,q$ are odd primes and $a,b \ge 2$

Now, clearly, $7^2 + 1 = 2\cdot5^2$. Is this the only solution? How would I prove this? Or if it is not the only solution, what would be the method to find other solutions? I'm not clear on how to ...
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Distribution of prime numbers. Can one find all prime numbers?

I want to know if it is possible to find a formula that gives all the prime numbers? or can one find the distribution of prime numbers? I know that there is a set of ongoing research on prime ...
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prime powers modulo prime

I stumbled upon the following property: $n\equiv n^5\bmod 5$ for all $n\in\mathbb{Z}$, so out of I tried other (prime) numbers $n\equiv n^p\bmod p$. My question is whether this is true for all primes ...
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Proving that if $a,b > 1$, then $5^a - 3^b=16$ has only one solution with $a=2$ and $b=2$

This may be one of those problems that is easy to state but very hard to prove. I don't know. I have tried to show that there is only one solution but I have not made much progress. Here's what I ...
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Connections between prime numbers and geometry

This might be a little open-ended, but I was wondering: are there any natural connections between geometry and the prime numbers? Put differently, are there any specific topics in either field which ...
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Subset of prime numbers

Given a subset of prime numbers say $A$. It is given that for $p,q\in A$ we also must have $(pq+4)\in A$ . Show that $A=\phi$ My work so far: It is obvious that $2,3\notin A$ . because all the ...
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Does a solution exist where $p,q$ are odd primes and $p^a - q^b = p^c - q^d$ where $a > c > 1$ and $b > d > 1$

From my thinking so far, there is no solution. Is this an open question or is the answer well known? Here's my reasoning about this issue: If a solution exists, then: $$p^c(p^{a-c} - 1) = q^d(q^{b-...
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What is the 5000th happy prime number?

Im writing a program that finds the Nth happy prime number. I think it works, but to double check I want to compare what it returns for the 5000th happy prime number. The problem is, I dont know where ...
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On the sum of prime powers

Has anybody investigated the asymptotic growth rate of functions in the form of $$f(z,n)=\sum\limits_{p\le n}p^z$$ For $Re(z)\ge -1$. Of course $f(0,n)=\pi (n)$ has an ocean of research surrounding it,...
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Is the product of two primes ALWAYS a semiprime?

I know by definition, a semi-prime has factors that are prime numbers. But what I'm unsure of, is if there is ever a case where the product of two prime numbers results in number with factors OTHER ...
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A function about prime numbers

Is there a function defined like this? $p(x)=1$ if $x$ is a prime, $p(x)=0$ if $x$ isn't a prime. If there is, what is the symbol of it?
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Proof for the existence of primes not equal to $ap_\alpha +bp_\beta$ etc?

Is there a general proof to show that there exists prime numbers larger than $min(p_\alpha,p_\beta)$that are not equal to $ap_\alpha +bp_\beta$, given $p_\alpha,p_\beta\in\mathbb{P}-\left\{2\right\}$ ...
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My first proof employing strong induction / complete induction (very simple number theory). Please mark/grade. [duplicate]

What do you think about my first proof employing strong induction? What mark/grade would you give me? Theorem Every natural number greater than 1 is a product of one or more primes. Proof First, ...
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Inequiality on prime gaps

Is there a good inequality for prime gaps. Like $p_{k}-p_{k-1}\leq f(k)$ ? In other words is there a known upper bound for $p_{k}-p_{k-1}$?
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Proof of convergence of $L'\left(1,\chi\right)$

can someone give me a good reference for a clear proof of the convergence of $L'\left(1,\chi\right)$, $\chi$ real-valued, non-principal Dirichlet character? Thanks in advance.
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Sum of reciprocals of every nth prime

I'm looking for a proof that $\displaystyle\sum_{n\mathop=1}^{\infty}\frac{1}{p_{kn}}$ diverges, where $p_n$ denotes the $n$th prime number and $k$ is a natural number. I know the proof that $\...
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Linear Independence for different fields

I have a statement for a space over $R^n$: {x, y, z} is linearly ind. $\implies$ {x + y, x + z, y + z} is linearly independent Quick proof: a(x+y) + b(x+z) + c(y+z) = 0 $\implies$ (a+b)x + (...
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Why is $\left(\mathbb{Z}_{51}\right)^* \cong \mathbb{Z}_2 \times \mathbb{Z}_{16}$?

I have to show that $\left(\mathbb{Z}_{51}\right)^* \cong \mathbb{Z}_2 \times \mathbb{Z}_{16}$. I know that $\mathbb{Z}_{51}\cong\mathbb{Z}_3 \times \mathbb{Z}_{17}$ and that $(\mathbb{Z}_p)^*\cong \...
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Prime values of binomial

Does there always exist $x$, such that $x>b$ $x>a$ and $a+bx^n$ is prime? Of course, $a$, $b$ are given relatively prime numbers. I know that is true for n=1 in general, and I understand that it ...
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Numbers used for modeling impose constraints on the model

In modeling observation we use different numbers. Mostly either positive integers or rationals. Both impose constraints on the model description. Positive integers have exactly one minimal element (...
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Questions about the proof that every odd integer is the sum of 5 primes

In http://arxiv.org/pdf/1201.6656.pdf, Tao proved that all odd numbers greater than 1 are the sum of 1, 3, or 5 primes. In page 36-37, he uses the fact that for all $x > 1.1\times10^{10}$, there ...
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Show that $16$ is a perfect $8$th power modulo $p$ for any prime number $p$ [duplicate]

Show that $16$ is a $8th$ power $\mod{}$ $p$ for any prime number $p$. I have no idea how to approach this. I tried, $$a^8\equiv16\pmod{p}$$ $$(a^4+4)(a^4-4)\equiv 0 \pmod{p}$$ $$a^4 \equiv \pm4\...
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Why $x \le (1+\frac {\ln(x)} {\ln(2)})^{\pi(x)}$ imply $\pi(x)\ge \frac {\ln(x)} {2\ln\ln(x)}$ for $x \ge 8$?

Let $\pi(x) = |\{ p \le x : p \in P\}|$ denote the prime counting function $\pi:\mathbb R \rightarrow \mathbb N$ and $P$ the set of primes. The equality $$x \le \left\lfloor \prod_{p_i\le x} 1+\frac {...
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Prime number greater than n

Consider the follwing problem: Given $n$ (in binary) output a prime number $p \geq n$ (not necessarily the first prime number after $n$) Are there better techniques than the trivial one that scans $...
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N-th k-almost prime satisfying a certain condition

Let $G(n,k)$ be the n-th k-almost prime. Prove that for every for every $n \in N$ there exists infinitely many $k \in N$ satisfying $2*G(n,k) = G(n,k+1)$. Source: http://mishabucko.wordpress.com
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Prove that for all $a\in \mathbb{Z}$ and all primes $p$, $p^2$ does not divide $a^2-p$

What would be a method to start, or some can prove useful theorem for this problem Prove that for all $a\in \mathbb{Z}$ and all primes $p$, $p^2$ does not divide $a^2-p$
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Does this function describing super-primes converge?

The prime numbers are positive integers that have no multiplicative structure. One method for counting the number of primes contained in a positive integer is sieving. As an example, for the integer ${...