Problems & Puzzles: Conjectures

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."

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