I'm a new user and have only a college sophomore's understanding of mathematics, so please bear with me.
I was reading a book titled “The Simpsons and their Mathematical Secrets” in which the author briefly discusses Mersenne primes and perfect numbers. Among his statements were two that piqued my interest:
- Every perfect number is also a triangular number, and
- Every perfect number contains a Mersenne prime as one of its divisors.
After reading this, and how rare perfect numbers are, I wanted to try and devise a method of generating perfect numbers from Mersenne primes. I thought the best way to do this was to take the sum of a set of consecutive integers which included the Mersenne prime. I hand tested starting with small Mersenne primes:
$$3 + 2 + 1 = 6$$ $$7 + 6 + 5 + … + 2 + 1 = 28$$
And using Gauss’s formula for larger ones:
$$31 \cdot \frac{31+1}{2}=496$$
I had expected to have to add greater consecutive integers to the sum in order to arrive the Mersenne prime’s corresponding perfect number, but after doing this method a few times It seemed like the Mersenne prime was always the largest number in the series of consecutive integers, in other words the “base” of the triangular number.
My question is, is this true for all perfect numbers?
Also, the author made another statement the Mersenne primes can be generated using the formula:
$2^p-1$ where $p$ is any prime number
But the author also stated that this formula does not always generate a prime number. If the number of primes are infinite, Is the problem with finding new perfect numbers that the equation above produced less and less Mersenne primes as larger and larger known primes were plugged into it?
Thanks for taking the time to read this.