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.

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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.

5 comments:

JasonZimba said...

Indeed, 38 is the *smallest* integer expressible as the sum of a prime and a perfect square in four different ways.

I won't be able to say this about my age again until I am 62 = 1^2 + 61 = 3^2 + 53 = 5^2 + 37 = 7^2 + 13.

When my dad has his birthday next year, he will attain the smallest age decomposable in *five* ways: 83 = 2^2 + 79 = 4^2 + 67 = 6^2 + 47 = 8^2 + 19 = 9^2 + 2.

JasonZimba said...

So, I have decided: (1) There are infinitely many integers that can be written as a sum of a square and a prime; and (2) there are infinitely many integers than cannot be written as a sum of a square and a prime.

For (1), it suffices to note that there are infinitely many squares that can be written as a sum of a square and a prime. For there are infinitely many odd primes p, and for each odd prime p, we have ((p+1)/2)^2 = ((p-1)/2)^2 + p.

For (2), it suffices to note that there are infinitely many squares that cannot be written as a sum of a square and a prime. For let q be any odd composite number, and put n = (q+1)/2. Then n^2 cannot be written as the sum of a square and a prime, for otherwise we would have n^2 = b^2 + p, whence (n+b)(n-b) = p, whence n-b=1, whence p = 2n-1 = q, contradicting primality of p.

JasonZimba said...

A fourth power cannot be written as a fourth power plus a prime. This is to say that the difference of two fourth powers cannot be prime. For if a^4 - b^4 = p, then (a^2 + b^2)(a^2 - b^2) = p. This implies a^2 - b^2 = 1. But this is impossible, as two squares cannot differ by 1. (The smallest possible difference between squares is 3, obtained with 1^2 and 2^2.)

More generally, the difference of two even powers a^(2n) - b^(2n) is never prime when n > 1. For if a^(2n) - b^(2n) = p, then (a^n + b^n)(a^n - b^n) = p. Therefore a^n - b^n = 1. But this is impossible, as two such powers cannot differ by 1.

Finally, one more piece of trivia about the number of the day: 38 is the smallest even integer not expressible as the sum of two odd composite integers. (We proved this in my X class last year - but not with my birthday in mind...!)

danimal said...

Had a chance to read the American Scientist piece on modern cosmology over lunch today. Very thought-provoking, and the plot of number of free parameters vs. independent observations is important to keep in mind in the spirit of healthy skepticism. I imagine a good number of the free parameters today have to do with the dark matter and dark energy, since we don't know what these are. Still, I think the theorists deserve a few more years, especially to look at results from the Large Hadron Collider at CERN which will start producing data next year sometime. I've read that some candidates for the dark matter may be able to be evaluated using this data, although I can't remember specifics.

It seems hard to get around the 'problem of initial conditions' in cosmology. We imagine that physical laws (general relativity, classical physics) should allow us to figure out past states of the universe from its present state. Given this, the options seem to be:

(1) The universe grew from a singularity, where the laws of physics as we know them break down.
(2) The universe has been around infinitely long, evolving according to the laws we (mostly) know.
(3) The universe materialized from a vacuum, in an initial state of very large size with specific arrangement of stars, planets (all matter) and the laws of physics took over immediately, and evolved it to where it is now.

The last option seems like an argument used by Creationists (which can't be disproved logically, but which blocks any further questions, questions which have led to good and self-consistent science), that the world was created 6000 years ago with a fossil/climate record which appears to go back billions of years.

But maybe option (2) should be explored more, based on the American Scientist article.

Last (speculative) thought: In quantum mechanics of simple systems, the collapse of the wavefunction prevents you from deducing past states from present states, as information gets lost in the "measurement". Although classical physics (including relativity) seems to hold sway over very large distances, maybe we don't completely understand how the universe as a whole "decoheres" (how classical behavior emerges for the universe as a whole). At least around black hole singularities, I have read that quantum effects ("quantum gravity") are important. Maybe this somehow makes our extrapolations back into the past, using relativity and classical physics, less certain than we think we are. If so, it could be harder to gather evidence to establish an accurate cosmology than we think. (I should go back and read "The Emperor's New Mind" by Penrose, I remember him talking a good deal about black holes and wavefunction collapse.)

JasonZimba said...

Danimal, I know I'm very slow and plodding about all this - I always have been - but I confess I'm not really sure that wavefunctions *do* collapse.