mysteriousprime

Mysterious Prime

(Truely Amazing)

Back to Main Page

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 P1 & P2 are prime numbers then:

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

Q.E.D

Page updated

Google Sites

Report abuse