mfb said:
You could share what you learned! The
Catalan–Mersenne number conjecture, I guess.
Where studying the MC conjecture took me today. To begin near the end:
In the set of positive integers > 5 there are no odd perfect numbers at least lower than 2
64 and probably for the entire set. Unproven but logical conjecture?
The search for new prime numbers using very large powers of 2 leads to and is very closely coupled to finding the next Mersenne prime. IOW it is computationally more efficient to calculate large powers of 2 and subtract 1 and then test the result for primality; i.e., show no factors other than 1 and the number, than it is to employ sieve algorithms with or without prior factorization.
Or in simpler terms: look for new primes among large Mersenne numbers (1 less than large power of 2) 2
n-1.
I am predisposed to look for twin primes but have not connected double-Mersenne numbers to that quest (if there is any known relation).
I was distracted/diverted from studying the MC conjecture by the set of perfect numbers. Unless I am being fooled by the (humorous)
Strong Law of Small Numbers, perfect number 28 is particularly useful in certain iterative applications. Most likely any connection to Lunar cycles is arbitrary and pre-scientific but its relation to biology, specifically perception, might be profound.
Need to think about what I read today and will relate better information tomorrow. Thanks.