Thread: Formula Development

Hello, Science Community or, in this case, those with specific Mathematical abilities. This is not a homework question, as I will state in my disclaimer here, it is merely something I wish to present to get your mathematical juices flowing.

I have been working on a project for the past little while where I am attempted to understand certain things about the lovely mystery that is Perfect Numbers and their counterpart, Mersenne Primes.

I had been progressing well until I came across a bump in the road.

I had an equation developed:

[2(2^p-1-1)+1(2^p-1)

This equation works when p is a Mersenne Prime. However, 2^p-1 is the only equation that develops Mersenne Primes, as you may notice I have incorporated into my equation.

2^p-1 doesn't work every time, though.

2^1-1 = 1
2^2-1 = 3
2^3-1 = 7
2^4-1 = 15 But 15 isn't Prime!

Well, there's the problem. Mersenne Primes may be used to develop a functional equation for Perfect Numbers but no consecutively functional equation has been developed for Mersenne Primes.

Therein lies the problem.

The first Mersenne Primes are:

1
3
7
31
127
8191
131071
524287
2147483647

For a more comprehensive list, go here: http://www.geocities.com/CapeCanaver.../ListOfMP.html

However, my equation works when the following numbers are substituted for p:

2
3
5
7
13
17
31
67

These numbers are the p in 2^p-1

What I am trying to do is develop an equation that produces these numbers consecutively with a specific rule. Well, I know that there aren't any equations out there to find these numbers and I know that many have tried, none have succeeded. But, if you want a challenge, as I always do, entertain me with your ideas.
Thanks.

~ Skiyk