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Mysterious Primes

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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& Pare prime numbers then:

(P1^P2+P2^ P1) MOD P1*P= P1+ P2

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)


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

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