# Tagged Questions

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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### 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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### 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
### 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 $a^2-p$