# Number Theory

Okay I hope it's okay if I have a couple question. I've been strugelling a bit with this problem set. About a quarter of the questions I just don't seem to see how to start them. Any hints would be greatly appreciated. Thank you kindly
I

## Homework Statement

Give that $p\nmid n$ for all primes $p\leq \sqrtn$ show that n> is either prime or the product of two primes.

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## The Attempt at a Solution

I don't really see how to start this one. Any hint would be greatly appreciated

II.

## Homework Statement

Give another proof of the infinitude of primes by assuming that there are only finitely many primes say $p_1, p_2, ... p_n$, and using the following integer to arrive at a a contradiciton:
N = $$p_2p_3...p_n + p_1p_3...p_n +...+p_1p_2...p_{n-1}$$

## The Attempt at a Solution

I think that this proof should involve showing that $p_k\nmid N\forall k$ so N must be prime. Which would be like like Euler proof, but I can't seem to see how to set that up

## Answers and Replies

Dick
Science Advisor
Homework Helper
For I), if all of the primes less than n^(1/3) don't divide n then n still has a prime factorization. How many factors can there be? For II) for any i, p_i divides all of the terms which sum to N except one. Can it divide N?