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    Prove that there exist infinitely many primes p such that p=3 mod 4.

    (Original post by DB9)
    Prove that there exist infinitely many primes p such that p=3 mod 4.
    first note if p1=4k+1 p2=4m+1 then p1p2 is of form 4n+1

    Assume p1,....pn are all the primes of the form 4k+3
    let P=p1p2....pn and consider the number
    since this is of the form 4P+3 by using the first observation (ie product of primes of form 4k+1 gives a number of that form) N must have a prime factor of the form 4k+3 for some k.
    But this prime factor can not be one the primes in our assumption since we would have a remainder of 3
    This is the required contradiction.
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Updated: October 10, 2006

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