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.

## Even numbers and twin prime

### 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.

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.

### 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.

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.

### Re: Even numbers and twin prime

It was easy to see the link with Goldbach conjecture.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.

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)