Re: an extension of Goldbach's conjecture
- From: Joe Blow <justarandomid99@xxxxxxxxx>
- Date: Thu, 06 Dec 2007 02:39:46 EST
To your knowledge, do any of you know if someone has
already claimed
the following conjecture:
"Any natural number x different from m where m
divides x can be
expressed as the sum of m primes."
I'm trying to find an honors project, and this is
something I've been
working on for a while. I can prove it if I assume
Goldbach's
conjecture...which of course doesn't prove it. But
since most
mathematicians are convinced of GC, they should be
convinced of mine
as well.
If you assume GC, your statement follows readily. There are three cases: (1) m is even, (2) m>3 is odd, (3) m=3:
(1) If m is even and x>=4, then 2*x/m is even and therefore expressible as the sum of two primes p1 and p2 (by GC). Then x = m/2*(p1+p2) is the sum of m primes.
(2) If m>3 is odd, then x-3*x/m >= 4 and is divisible by the even number m-3, and hence x-3*x/m can be expressed as the sum of m-3 primes by the result in (1). However, by GC, 3*x/m by can be expressed as the sum of 3 primes. Thus, x is the sum of m primes.
(3) If m=3 the statement is implied by GC.
.
- References:
- an extension of Goldbach's conjecture
- From: Chase
- an extension of Goldbach's conjecture
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