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Number Theory

  1. Jan 22, 2008 #1
    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
    1. The problem statement, all variables and given/known data

    Give that [itex]p\nmid n[/itex] for all primes [itex]p\leq \sqrt[3]n[/itex] show that n> is either prime or the product of two primes.

    2. Relevant equations

    ?

    3. The attempt at a solution

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

    II.
    1. The problem statement, all variables and given/known data

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

    2. Relevant equations



    3. The attempt at a solution

    I think that this proof should involve showing that [itex]p_k\nmid N\forall k[/itex] so N must be prime. Which would be like like Euler proof, but I can't seem to see how to set that up
     
  2. jcsd
  3. Jan 22, 2008 #2

    Dick

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    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?
     
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