## Even numbers and twin prime

Primes, divisors, arithmetic, number properties, ...
Alhazen
Posts: 94
Joined: Sun Feb 20, 2011 9:55 pm

### Even numbers and twin prime

Hi,

Let p,q,q+2 be odd prime numbers
q and q+2 are twin prime 2n = p*φ(q+2)-q*φ(p) where φ is Euler totient

What are the even numbers that we can not write in the form above ?

Thank you for any clue.
drwhat
Posts: 42
Joined: Tue Sep 06, 2011 4:56 am

### Re: Even numbers and twin prime

Well since for any prime p, φ(p) = p-1 your equation reduces to
2n =p*(q+1) - q*(p-1)
2n = pq+p-pq-q

2n = 2pq + (p-q) where p is any odd prime, and q is the smaller of a twin prime pair.

I'm not sure what else we could say about n in this situation to elimnate possible solutions.
rayfil
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### Re: Even numbers and twin prime

2n =p*(q+1) - q*(p-1) = pq + p - pq + q = p+q

2 and 4 are the two even numbers which cannot satisfy the equation if p is allowed to be the same prime as q or q+2. Otherwise, you would have a few more even numbers.
When you assume something, you risk being wrong half the time.
Alhazen
Posts: 94
Joined: Sun Feb 20, 2011 9:55 pm

### Re: Even numbers and twin prime

rayfil wrote:2n =p*(q+1) - q*(p-1) = pq + p - pq + q = p+q

2 and 4 are the two even numbers which cannot satisfy the equation if p is allowed to be the same prime as q or q+2. Otherwise, you would have a few more even numbers.
It was easy to see the link with Goldbach conjecture.
The idea was to link the statement above with this statement :

Any m=ODD number>1 can be expressed at least once as :
m=pq-φ(pq) with p and q odd prime

We can easily prove that
m+2=p*(q-2)-φ(p*(q-2)) with p,(q-2) odd prime and q-2 the first number of twin prime.

By mathematical induction we can show that any ODD number can be written as
m=pq-φ(pq)