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Prime proofs

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1. If and are distinct primes prove that for any integer a,
Use Fermat's theorem

2. Show that if and are both primes, then 4[ (mod
Use Wilson's theorem.

3. Let be an odd prime. Prove that if g is primitive root modulo and (mod is not

Use the binomial expansion

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Solution Summary

There are three proofs here that use Fermat's theorem, Wilson's theorem, and binomial expansion.

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1. Proof:
Since and are distinct primes, then . According to Fermat's Theorem, for any integer , we have (mod ) and (mod ).
Now we check the expression .
Since (mod ) and (mod ), then we have and . Thus .
Similarly, (mod ) and (mod ), then ...

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