Monday, September 17, 2007

Rochester Turning

Just a heads-up to those who are interested...

Today I heard from a friend who is writing for a local political blog called Rochester Turning. Admittedly, I'm not a technophile by any means, but this kind of site, a group blog, was something new to me. Because the blog has so many writers, it's updated constantly - much more often than a single-person blog. The blog is about local politics in the upstate-NY area. Essentially, it seems to me that they are using the weblog technology to create, at very low cost, a burgeoning news site that bypasses the conventional media. Some of their press is here.

It's also interesting to see how intensely the writers have thrown themselves into the work. These are professionals and busy people. But many of the posts say things like, "...I'm here at the convention with X and Y...", and, "...I took a long lunch to hear the speech by Z..." This is not armchair stuff. These people are putting some significant hours into learning their landscape, as a way of doing something to take back this country.

I'm glad to see it. And I'm wondering what more I myself can do. We have left this sort of work to the worst of us for too long.

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...!

Saturday, September 1, 2007

The Prime of Life

In my post Where Credit Is Due, I posed the question, Can somebody tell me when I became middle-aged? Well, the answer is, "today." Today I'm 38 years old - and according to the actuarial table here, 38 is the very age when a man's present age equals his expected remaining years of life. However, a bit of linear interpolation on the data suggests that I've got a little time left. I won't pass the halfway point of the table until June 7th of next year, around dinnertime. No need to rush into that motorcycle purchase just yet.

I also made this graph using the data from the actuarial table. It shows the risk of dying in the next year for men of different ages, from age 10 to age 50. Some interesting patterns....



(Before anybody panics, let me point out that the vertical scale only goes up to 1%.)

It so happens there's a piece on mortality statistics in this month's American Scientist - one of the best magazines of any kind published today. The article is here, in the Marginalia section.

***

A couple of random notes, and then I've got some Key Lime Pie to attend to.

(1) In that same issue of American Scientist, there is a courageous article on the scandalous state of modern cosmology - and by extension, the deep confusion within contemporary theoretical physics as a whole. The article is "Modern Cosmology: Science or Folktale?".

(2) A friend has generously mailed me a copy of The Black Swan: The Impact of the Highly Improbable. I'll give it a more serious look soon, but on first flip-through, I have to say, it should have been called The Black Swan: The Impact of the Highly Unreadable. Somebody get this guy an editor! Better yet, give the whole thing to Malcom Gladwell, and let him condense it down to a nice pithy piece for the New Yorker. Having said that, the bibliography looks very valuable, and the graph on page 276 is immediately convincing. There is obviously a lot here, although the author's pretentious style keeps him in the foreground, at the expense of his message.

(3) Today it struck me that 38 can be written in four different ways as the sum of a prime and a perfect square: 38 = 1^2 + 37 = 3^2 + 29 = 5^2 + 13 = 6^2 + 2. Amazing! In number theory, you frequently see decompositions into squares and decompositions into primes, but I have never seen a mixed decomposition problem like this.