**Conjecture 57. |P(i-1)-2P(i)+P(i+1)|
**

Mahdi Meisami sends the following
conjecture:

Every even number is of the form
|P(i-1)-2P(i)+P(i+1)|

P(i) is the i-th prime number.

Contributions came from Farideh
Firoozbakht & Luis Rodríguez:

***

Farideh wrote:

I think by omitting absolute value
in this conjecture we come to a more

interesting and stronger conjecture that we can state it in this way.

Every even number is of the form P(i-1)-2P(i)+P(i+1) where P(i) is

the i-th prime number.

This conjecture is a special case of the following nice conjecture.

If a, b are two natural numbers such that gcd(a,b)=1 then every

even number is of the form

a*P(i-1)-(a+b)*P(i)+b*P(i+1) where P(i) is the i-th prime number.

***

Luis wrote:

This conjecture follows from the
known conjecture:

"Any even number must be a difference between consecutive primes, and
that difference is infinitely repeated."

Demonstration:

Let be p = P(i) ; p - k = P(i-1) ; p + j = P(i+1)

Then 2N = 2p + j - k - 2p = j - k

That is : "Any even number is equal to the difference between certain
prime and its anterior, subtracted from the diference between the same
prime and its posterior."

***