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Demostrate:

If the number $M_p=2^p-1$ is Composite number, where $p$ is prime, then $M_p$ is a Pseudoprime.

This exercise was on a test and I could not do!!

Number Pseudoprime: Fermat's little theorem states that if p is prime and a is coprime to p, then ap−1 − 1 is divisible by p. For an integer a > 1, if a composite integer x divides ax−1 − 1, then x is called a Fermat pseudoprime to base a. It follows that if x is a Fermat pseudoprime to base a, then x is coprime to a. Some sources use variations of this definition, for example to only allow odd numbers to be pseudoprimes

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  • $\begingroup$ what is a pseudo prime? $\endgroup$ – mookid Nov 27 '14 at 16:44
  • $\begingroup$ ckeck it: en.wikipedia.org/wiki/Pseudoprime $\endgroup$ – Yobani Ordoñez Nov 27 '14 at 16:45
  • $\begingroup$ There are many different definitions of pseudoprime, as you can see in your Wikipedia link. Which one are you working with? $\endgroup$ – Brandon Carter Nov 27 '14 at 16:47
  • $\begingroup$ o i didnt know,, im sorry!!! $\endgroup$ – Yobani Ordoñez Nov 27 '14 at 16:49
  • $\begingroup$ It is pseudoprime to the base $2$. $\endgroup$ – André Nicolas Nov 27 '14 at 16:50
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More precisely, $M_p$ is a pseudoprime to the base $2$. To show this we show that $$2^{M_p-1}\equiv 1\pmod{M_p}.$$ By Fermat's Theorem we have $2^{p-1}\equiv 1\pmod{p}$. Thus $2^{p-1}=1+kp$ for some integer $k$, and therefore $M_p-1=2kp$. Thus $$2^{M_p-1}=(2^p)^{2k}=(1+M_p)^{2k}\equiv 1\pmod{M_p}.$$

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