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        riddles >> hard >> Prime pairs is NOT hard
        
(Message started by: Dmitriy on Jul 30^th, 2002, 12:42am)

Title: Prime pairs is NOT hard
Post by Dmitriy on Jul 30^th, 2002, 12:42am

Guys, this one is the simplest of them all.

1. Out of 3 successive numbers one can always be divided by 3. The proof is elementary.
2. Each second number is even.
3. Primes can only be divided by themselves.

Thus the number between them can be divided by both 2 and 3, which means it can be divided by 6.

Title: Re: Prime pairs is NOT hard
Post by Gamer555 on Jul 30^th, 2002, 9:10am

That's a good proof, lots simpler than the one I had.

What I would say is: (EXACTLY what you said, am not taking credit for what you said)

All primes above 6 odd. So the number between them is even, and divisible by *2*

Out of 3 numbers, one must be divisible by three, and since all prime numbers aren't divisible by three, the one in the middle must be.

Is this simpler to read?

Title: Re: Prime pairs is NOT hard
Post by NickH on Jul 30^th, 2002, 10:20am

The above are all good arguments.

An equivalent approach is to observe that all primes greater than 3 are of the form 6n-1 or 6n+1. (6n, 6n+2, 6n-2, 6n+3 are clearly composite.) Thus prime paris must be 6k-1 and 6k+1, for some k, and so the number in the middle is 6k.

Title: Re: Prime pairs is NOT hard
Post by anshil on Aug 1^st, 2002, 12:57am

Okay other question, how many prime pairs are there?

If you guess infinite, can you mathematically proof that there are infinite number of prime pairs?

( Similar like there exists a proof that there are infinite number of primes )

Title: Re: Prime pairs is NOT hard
Post by NickH on Aug 2^nd, 2002, 11:45am

As I recall, it is an open question whether the number of prime pairs is finite or infinite. It IS known that, whereas the sum of the reciprocals of the primes is infinite, the sum of the reciprocals of all prime pairs is finite. In fact, it is less than 2.