Power of prediction put to the test

If number theory means knowing what numbers will do then it has predictive power, right? Well there is an ancient equation which puts predictive power to the test.

Years ago I found myself at a critical point with the equation x² - Dy² = 1.

Some of my mathematical theory told me that there should be a large fundamental solution, the smallest positive integer solution with y not equal to 0, if D is prime and D-1 had 4 as a factor and small primes as factors, as long as neither D-1, D+1, D-2 nor D+2 is a square.

The most famous case is D=61, with a very large fundamental solution, which I already knew behaved as my research predicted:

61-1 = 60 = 4*3*5

And mathematical science is the art of prediction about the behavior of numbers. So in front of me was a simple test. What about 4*3*5*7 + 1?

I could just go to the next prime and see what happened. If my mathematical research was correct then the numbers had no choice. They would have to behave as predicted. But what if they didn't?

Well, if they didn't, I was wrong.

Turns out that 4*3*5*7 + 1 is 421 and I found the fundamental solution for D=421 on the web, with a quick search, why? Because it's a HUGE solution, so it was listed out there, by people who probably had no clue why it was so huge. Actually I'm sure they had no clue, any more than ancient people understood why they have weight before Sir Isaac Newton. But you see, they didn't have the mathematical science I had.

I felt relief.

If you're curious you can keep checking like check D = 4*3*3*5 + 1 = 181. Did some more web searching and found this useful table and yup, it's on it!

That's with just some of the rules of the mathematical predictive framework.

I put all the rules in a blog post:

somemath.blogspot.com/2011/09/cool-rules-renaming-pells-equation.html

My research tells you predictably how the fundamental solution will behave, so is mathematical science research. Science has a predictive power which people may fail to realize, but it gives predictive certainty.

Mathematical science gives predictive certainty about the behavior of numbers.

So mathematics is a science when it gives you predictive ability as that is a key component of science in general.

Now we can contrast with mathematical scholarship.

Mathematical scholarship has an emphasis on history and other facts of interest to scholars, so for instance you might learn of a guy named John Pell, and find that irrelevant to understanding the ancient equation. I don't care about such things. But my scientific perspective looks for predictive power.

Notice mathematical scholars can give you a way to solve the equation with integers, which does work. But ask them to explain why it works. (I think I can but the probable answer I have in mind is not very exciting to me.) Or also, ask them to tell you when solutions will be larger or not, which is the predictive power mathematical science offers.

This example is a great way to see how traditional mathematicians differ from scientists. Do a web search on Pell's Equation size, read through some published papers, and ask yourself, is that what mathematics is to you?

From those papers, it is clear that for a math scholar it isn't necessary for a mathematical technique to actually tell you much that is useful about how actual numbers should behave! Those papers reveal to me more of an exercise in style.

The scientific itch can drive discovery: what if mathematicians had been irritated by the lack of ability to do something as basic as figure out when the fundamental solution would be large or not?

Well then, the answer might not have been left out there for me to discover.

Predictive power can appear to tell you why, when it actually tells you what will happen. Knowing what will happen with electricity in circuits and through a filament can give you light at the flip of a switch.


James Harris