# Mysterious Primes

On analysis of prime numbers I found following few unique relationship which I couldn’t find anywhere else.

**Few Unique Relationship between two Prime Numbers:**

If both P_{1 }& P_{2 }are prime numbers then:

(P_{1}^P_{2}+P_{2}^ P_{1}) MOD P_{1}*P_{2 }= P_{1}+ P_{2}

I had posted this as a question on Yahoo Answers community & following is the proof given by Dannix.

Let P1=p and P2=q.

Proof: From Fermat’s Little Theorem we have

q^p ≡ q (mod p)

p^q ≡ p (mod q)

Also it is trivial that,

q^p ≡ q (mod q)

p^q ≡ p (mod p)

By the Chinese Remainder Theorem we get

q^p ≡ q (mod pq)

p^q ≡ p (mod pq)

Adding the two congruences yields

q^p + p^q ≡ q+p (mod pq)

Q.E.D

Link: http://answers.yahoo.com/question/index?qid=20081109033733AAwx5lh

Mysterious Primes,