prime number (a form like Mersenne primes)

Question

I found a form like Mersenne prime number and i wanted to be sure if its maybe better but i was wrong but still as good as Mersenne form its $(2^p+1)/3=P$ and p,P are primes P also can be a semiprime. I treied big numbers between 20 and 30 digits and i got good results but i could not check bigger number because i could not finde good website for that. So i want to ask if this form is it really like Mersenne or better or maybe wrong ? sorry my english is not that good .

Answer 1

Here's what I can give for information:

$$2^p-1=2kp+1\land k=3m\implies \frac{2^p+1}{3}=2mp+1$$ and by Sieve of Sundaram $mp\neq2ij+i+j, \forall i,j\geq 1\in\mathbb{Z}$ in order to be prime.

But in general, we don't say one form is better than another. Mersenne Primes are often the largest known primes, because more theory is easily applied in searching for them. While $2np+1$ is a form that is a magma(algebraic- structure)-like under multiplication, It's not completely closed off divisor wise.

Examples of what I mean ( and some that may not be)

• Mersennes are a divisibility sequence
• Mersennes are repunits
• Mersennes are 1 less than powers of 2 ( allows for fast mod, but generalizes)
• Mersennes have divisor form restrictions via Fermat's little theorem
• Mersennes have divisor form restrictions via Quadratic reciprocity
• Mersennes have prime divisor restrictions via a generalization of the Sieve of Sundaram
• Mersennes have a primality test (LL)
• Mersennes have a reduced primality test
• Mersennes have a PRP like primality test derived from the reduced version
• Mersennes have exponent restrictions related to Cunningham chains of first kind.
• Mersennes have restriction of divisors based on Polynomial remainder theorem