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Thread: Primes 2

  1. #1
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    Primes 2

    prove that there are infinitely many primes by considering the sequence 2^2(^1) +1 , 2^2(^2)+1 , 2^2(^3)+1 ...

    note the parantheses indicates the power of the first power.
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  2. #2
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    Quote Originally Posted by eyke View Post
    prove that there are infinitely many primes by considering the sequence 2^2(^1) +1 , 2^2(^2)+1 , 2^2(^3)+1 ...

    note the parantheses indicates the power of the first power.
    if n > m \geq 1, then 2^{2^n}+1=(2^{2^m})^{2^{n-m}}+1=(2^{2^m} +1 - 1)^{2^{n-m}}+1 \equiv 2 \mod 2^{2^m} + 1. so if d \mid 2^{2^n}+1 and d \mid 2^{2^m} + 1, then d \mid 2, which implies that d=1.

    so every two elements of your infinite sequence are coprime and each term has at least one prime factor. thus the number of primes must be infinite.
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