Driving over to the nursing home to visit my parents the other day, it occurred to me to change the word "sum" in Goldbach's conjecture to "difference." Here then is the conjecture:
Every even number is a difference of two primes.
Examples: 2 = 5 - 3 and 65,036 = 65,053 - 17.
I have verified the conjecture out to 65,036. For me to go beyond that would require a little more investment of time.
I also made bold to send Professor Oliveira e Silva the conjecture, and he very kindly answered, saying that it was not known whether this conjecture is true, but that it does appear to be so.
Stopping at Burger King to pick up some hamburgers for my folks, I jotted down on a napkin a more general problem: given integers M and N, for what integers K is it the case that K is a weighted average of primes,
K = (Mp + Nq)/(|M| + |N|)
for some primes p and q. With M = N > 0 we have Goldbach's conjecture. With M = -N, we have (OK, until I hear otherwise, let's just say it) "Zimba's conjecture." (Henceforth abbreviated ZC.)
Number theory is a valuable subject for an educator like myself, because some of the discipline's hardest questions are so near the surface. Or as the number theorist G.H. Hardy put it, "...there are theorems, like 'Goldbach's Theorem,' which have never been proved and which any fool could have guessed."
More notes, as I read up on this:
In 1849, Alphonse de Polignac conjectured that every even number is the difference of two consecutive primes. This has not been proven, but it would imply ZC if true. (Ref)
The thread continues in the comments below: a reference for the ZC, and another try at a conjecture - this one new for sure...!