Monday, September 10, 2007

Goldbach Variations

After my last post (see especially the comments), I got to thinking about prime numbers and prime decompositions. The most famous unsolved problem along these lines is Goldbach's Conjecture, which hypothesizes that every even integer greater than 2 is a sum of two primes. (Examples: 4 = 2 + 2, 18 = 7 + 11, etc.) Goldbach's conjecture has been verified for every even number up to 100,000,000,000,000,000 (10^17) by the Portuguese mathematician Tomás Oliveira e Silva and his research group; their results are here. But nobody knows whether the conjecture is true or not. There could be a counterexample just around the corner.

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...!


danimal said...

You inspired me to read a little about prime numbers on Wikipedia after reading your post. It definitely makes thinking about prime numbers enjoyable (and tantalizing) that some of these conjectures, which are simple to state and fiddle with, aren't yet proven or disproven. It's also nice to hear about the gracious attitude of Prof. Oliveira e Silva in answering your email.

Wikipedia seems to have something like the ZC listed under a "weaker form of the de Polignac conjecture" on their main page on "Prime Numbers". But if the "weaker" form is not yet proven, it would seem to deserve its own name, and the "ZC" fits the bill very nicely!

Note that the Polignac conjecture (as I read it) actually states that every even number can be written as the difference of two consecutive primes an **infinite** number of different ways/times (I'm sure you noticed this too). The flavor of prime number conjectures/theorems often seems to be that if something is true once, it is true an infinite number of times. It could be interesting to try to prove that if the ZC is true, it is also true for each even number an infinite number of times.

I monkeyed around for a couple of hours with the Goldbach conjecture and was surprised to find that for even numbers 1 to 100 it is abundantly satisified. Roughly, even numbers from 1-100 were expressible on average in 4 different ways, with the highest being 90 with 9 different ways. Also, the number of expressible ways rose slowly (though irregularly) with the even number considered. I wonder if this trend continues past 100... If you think of the whole set of prime numbers being "pasted" onto the number line starting at 3, and then again starting at 5, and again starting at 7, it seems as if the number "hits" at the even numbers would get denser and denser.

I also tried out one method of a proof. Wish I could say that "I found a proof, but this post is too small to contain it" :), but as expected I found a counterexample to the idea. I'll try to post a link to a graph and my tinkering idea sometime this week for fun, though.

JasonZimba said...

Danimal, thanks for the pointer to this page.

So, ZC is old news; so let's try out another one. This arises from the "KMN," weighted-average formulation of the problem:

Every odd multiple of 3 can be written in the form 2p - q, where p and q are prime.

I verified this up to 75621 = 3*25207 = 2*37813 - 5.