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Math Help - Primes above googol

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    Primes above googol

    Prove that 10^100+1 is not the smallest prime greater than googol.
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    Quote Originally Posted by meshel88 View Post
    Prove that 10^100+1 is not the smallest prime greater than googol.
    You can prove that 101 divides 10^n+1 when n\equiv 2\pmod{4}... You just need to find what the other factor is. But this doesn't answer the question... cf. Soroban's post.
    Last edited by Laurent; April 1st 2010 at 11:42 AM. Reason: shift in modulo (n=2 is obvious...) ; cf. chiph588
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    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by Laurent View Post
    You can prove that 101 divides 10^n+1 when n is divisible by 4... You just need to find what the other factor is.
     101 \not|10^n+1 when  4|n

     101 \mid10^n+1 when  n\equiv 2\mod{4}

    Try it out with small  n to see for yourself.
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    Hello, meshel88!

    Prove that 10^{100}+1 is not the smallest prime greater than googol.

    Fact: . 10^{100}+1 is a composite.


    Proof: . 10^{100}+1 \;=\;\left(10^{20}\right)^5 + 1^5 \;=\; \left(10^{20}+1\right)\left(10^{80} - 10^{60} + 10^{40} - 10^{20} + 1\right)

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