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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diophantine equation $ |x^2-py^2|=\frac{p-1}{2} $

Prime $p\equiv3\pmod4$, then the equation $$ |x^2-py^2|=\frac{p-1}{2} $$ has a solution in integers obviuosly, $ x^2-py^2 = -1 $ has no solution in integers. How to attack this problem? Thanks ...
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About primes and counting them. [on hold]

Are there bounds to the prime counting function that do not involve logarithms?
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1answer
44 views

Riemann Zeta circularity?

In this post I show: $$\prod _{p\text{ prime}} \frac{p^s}{p^s-1} = \zeta(s).$$ Wolfram Alpha shows an alternate form for the primes: $$\frac{p_n{}^2}{p_n{}^2-1}=\frac{\left(\sum _{k=1}^{2^n} ...
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A prime connection between two numbers with same prefix

If I know that the number n is prime, is there a fast algorithm to check if 10*n+k is prime, where k is 1, 3, 7 or 9? I mean, an algorithm based on the fact that n is prime. Thanks for help! P.S. : ...
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Question on the Prime Number Theorem (the Tchebychev Function)

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
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Is the following statement true

Is the following statement true and how to prove it? \begin{align} (a^2)^{3N} \equiv a^2 \mod{p} \end{align}
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solutions to linear equations involving prime numbers?

Suppose we have the two equations: $2Z - p = Xq$ $2Z - q = Yp$ where $X,Y,Z \in \mathbb{N} $ and $p,q \in \mathbb{P} - \left\{2\right\} $ Are there any solutions where $Z$ isn't prime?
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1answer
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How does this algorithm find the largest prime factor?

This question on math.stackexchange details an algorithm that can be used to find the largest prime factor of a number. I used it to solve Project Euler #3. Here's a short description of the ...
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1answer
53 views

Coprimality of $2^n + 3^n$ and $5^n + 7^n$

Prove or disprove that for all positive integers $n$, $2^n +3^n$ and $5^n + 7^n$ are always coprime. This is not a homework problem, it is merely a problem I set myself after doing some number theory ...
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1answer
32 views

Is there a asymptotic formula for product of primes? [duplicate]

$$P(x)=\prod_{p\leq x}p$$ As you can see P(x) represents the product of primes which are not greater than x. Is there a asymptotic formula for this?
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1answer
22 views

Dirichlet prime counting function?

Let $a$ and $b$ be coprime (i.e. $a \perp b$). Let $f(a,b,x)$ denotes the number of the primes such that $p=ak+b$ and not greater than $x$. For example $f(4,1,10)= 1$. Is there an asymptotic formula ...
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Firoozbakht's conjecture solution?

Not so much an question as adding another level to the same question as Ratio of logarithmic primes. (See answers, same as here.) The Firoozbakht's conjecture (1982) is equal to: $$(p_{n+1})^{n} ...
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1answer
44 views

infinite primes $p\equiv1\pmod n$ without cyclotomic polynomial

Without cyclotomic polynomial, is there an elementary proof of the following: for each integer $n>1$, there are infinitely many primes $p$ such that $p\equiv1\pmod n$ ? please don't refer to ...
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1answer
48 views

Explain theorem in Number theory

can some one explain with a clear example this theorem for me, Let ($A_1$, $A_2$, $A_3$,..., $A_n$) be integars and $p$ a prime number. if $p|(A_1A_2A_3...A_n)$ then there exist some $1 \leq k \leq ...
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1answer
33 views

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 N > 230 such that the number of ...
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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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1answer
23 views

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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1answer
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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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1answer
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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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73 views

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, ...
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1answer
38 views

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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2answers
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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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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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1answer
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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 < ...
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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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Find the value of difference between ${x}$ and $[x]$ [closed]

[x] = Greatest prime number less than x and {x} = Smallest prime number greater than x. Then {231} - [231] = ? Is there some trick to solve this problem?
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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 this ...
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1answer
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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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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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1answer
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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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Primes probability for $2^{2(ak+b)}-3$

I'm working on the following problem: If $x$ is a prime and of the form $ak+b$, is there a possibility to check, whenever $2^{2x}-3$ could be a prime or not, without calculating it or extracting ...
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183 views

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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If $p^a \equiv -1 \pmod {q^b}$, is there anything that we can say about $a$ if $p,q$ are odd primes and $a,b > 1$

If $p^a \equiv 1 \pmod {q^b}$, then, from Carmichael's Theorem, we know that: $a = u\varphi(q^b) = u(q-1)(q^{b-1})$ where $u \ge 1$ Can we say anything similar if $p^a \equiv -1 \pmod {q^b}$
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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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primes like sophie germain primes

Show that if there exist infinitely many positive integers that cannot be represented as $xy+yz+zx$ for any natural $x,y,z$, then there exist infinitely many primes $p$ such that $2p-3$ is also prime. ...
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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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1answer
58 views

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) = ...
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Find $a,b,c \ge 2$ and $p,q$ odd primes where $p^a - 1 = c*q^b$

I've been recently thinking about finding primes $p,q$ where the power of one divides the power of the other when subtracted by $1$. For example, if we remove the requirement that $p,q$ be odd ...
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1answer
58 views

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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Miller-Rabin primality test and testing one

I'm learning about Miller-Rabin primality test but in all the problems I see in the notes of a person I got them from, I see that even if he expressed the number as $2^1 \cdot something$, he still ...
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1answer
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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 ...
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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 ...