This is something I've been toying with the last few days.
I can prove that all even positive integers except 2, 4 and 8 can be written
as the sum of Mersenne primes.
All above 21 are either 0, 1 or 2 mod 3, and are therefore the sum of either
1, 2 or 3 sevens with a sufficient number of 3's thrown on top.
I am sure this can (and should) be stated far more formerly, but my real
question is this: Is it possible to strengthen this conjecture, say by
putting a ceiling on the number of times that any one prime need be
repeated?
Such a statement can also be made for the odd positive integers. 1, 5 and
11 are the only exceptions that need be made, since all odds above 13 can be
written as the sum of an even number above 8 and a Mersenne prime.
To my knowledge, this line of inquiry is original with me. However, if
anyone can think of any work in this direction, please by all means let me
know.
Regards,
Nathan Russell
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