Fermat's Room; Goldbach's Conjecture.

I reached my peak with numbers when I was a child at P. S. 217 (under the tutelage of Mrs. McNulty).  Addition, subtraction, multiplication, division -- that's my area of expertise. Geometry, algebra, trig -- way too abstract for my pedestrian brain. Calculus and up -- not the least chance.

Arithmetic yes, mathematics no.

Sometimes I try to read popularizations of mathematical theory, but I rarely make it beyond the first few sentences. Once symbols start to appear, I'm done. Especially Greek letters (in parentheses). 

The consolation prize is that I'm better than most people at mental arithmetic. I can do sums and approximations and other stuff in my head.

Last night, we watched Fermat's Room. It's the kind of movie that is often called an "ingenious thriller."  It's also a member of a very small group of films that concern mathematics or mathematicians:  Good Will Hunting, Stand and Deliver, Antonia's Line, Straw Dogs. 

Fermat's Room focuses on a possible proof of Christian Goldbach's Conjecture. I suppose that I should have known about Herr Goldbach and his guess, but I confess to ignorance. Goldbach's conjecture dates to 1742 and is apparently one of the oldest unsolved problems in number theory. In brief, Goldbach theorized that every even integer greater than 2 can be written as the sum of two prime numbers. For example, 10 is equal to 3 plus 7, 20 to 17 plus 3 (or 13 plus 7), and 100 resolves into 47 plus 53. Although no one has found a number that doesn't work, the conjecture can't be proved -- or, at least, hasn't been proved because there are some very large numbers out there. In Fermat's Room, two of the characters claim to have proved the conjecture. And they are consequently at odds.

I had always thought of prime numbers as a kind of novelty or curiosity. But if primes are a component of all even numbers -- well, that's something.  Goldbach's idea raises the stature of primes in my perennially naive eyes -- in exactly the same way that the Fibonacci series became more important to me when I learned that the chambered nautilis is structured Fibonacci-wise.  

While watching the movie (which is a good one), I ran through the even numbers from 10 to 70.  Everything was in order. But why?

Later in the evening, I read that every even number can also be resolved into the sum of a semi-prime (which is the product of two primes) and a prime. So 100 equals 77 (7 time 11) and 23. I spent a restless  night summing, in my bed, semi-primes and primes from 10 to 100. All in order, once again.

Why should it be so?

Whether it should be, or not, it is.  I remember that I used to answer my children and my students with the response, "That's not a why question?" Goldbach's Conjecture is a fact, not a why.

It's like gravity. I understand that the apple falls to the ground and that the moon doesn't head out into space and that it's harder to walk uphill than downhill. But I still don't understand why. It works, and that's enough. For me.

Comments

[Huphoz]

Mathematics is the queen of sciences and Arithmetic the queen of mathematics. Hence aptness for arithmetic puts one in the exclusive club of the likes of Euler, Pascal and Gauss. Or to put it another way Arithmetic is Number Theory.