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Math Help - Congruence with FLT

  1. #1
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    Congruence with FLT

    The congruence 2^{52532} \equiv 1 \bmod{52633} is true. Can you conclude that 52633 is a prime number?

    So would the answer be that since 2^{52532} \equiv 1 \bmod{52633} is in the format for FLT then 52633 must be prime right?
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  2. #2
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    Quote Originally Posted by SCRandom View Post
    The congruence 2^{52532} \equiv 1 \bmod{52633} is true. Can you conclude that 52633 is a prime number?


    Of course not: why would anyone conclude such a thing?? Of course, it's pretty easy to check directly that 7\mid 52633 , but even without this

    we couldn't conclude that: for example, 2^4=\!\!\!\pmod{15}...can we conclude 15 is prime? 2^6=1\!\!\!\pmod 9...is 9 a prime?

    About what you wrote below: " ...is in the format of FLT..." do you mean Fermat's Little Theorem? What is in this format??

    Tonio


    So would the answer be that since 2^{52532} \equiv 1 \bmod{52633} is in the format for FLT then 52633 must be prime right?
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by SCRandom View Post
    The congruence 2^{52532} \equiv 1 \bmod{52633} is true. Can you conclude that 52633 is a prime number?

    So would the answer be that since 2^{52532} \equiv 1 \bmod{52633} is in the format for FLT then 52633 must be prime right?
    I think you meant to say 2^{52632} \equiv 1 \bmod{52633} , as 2^{52532} \not\equiv 1 \bmod{52633}.

    This doesn't mean  52633 is prime. flt says if  p is prime and  (a,p)=1 , then  a^{p-1}\equiv 1\bmod{p} . The converse however is not true.

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