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Thread: Congruence with FLT

  1. #1
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    Congruence with FLT

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

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


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

    we couldn't conclude that: for example, $\displaystyle 2^4=\!\!\!\pmod{15}$...can we conclude 15 is prime? $\displaystyle 2^6=1\!\!\!\pmod 9$...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 $\displaystyle 2^{52532} \equiv 1 \bmod{52633}$ is in the format for FLT then $\displaystyle 52633$ must be prime right?
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  3. #3
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    Quote Originally Posted by SCRandom View Post
    The congruence $\displaystyle 2^{52532} \equiv 1 \bmod{52633}$ is true. Can you conclude that $\displaystyle 52633$ is a prime number?

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

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

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