Saturday, April 27, 2013

Without explanation

One of my more dramatic accomplishments was providing the first explanation in history for the size of the smallest non-zero positive integer solution to x2 - Dy2 = 1.

The equation has been known for a while, and is commonly called "Pell's Equation" by mathematicians.

It gained a more recent burst of fame--it is an ancient equation with a long history--from Fermat who used it for an intellectual challenge--math as a game.

However, it is unlikely even the great Fermat understood why his particular solution was as big as it was.

What he could use--as the solution was HUGE by the standards of the 17th century--was D=61.

Here the smallest nonzero integers x and y that can possibly work are:

x = 1766319049 and y = 226153980.

17663190492 - 61(226153980)2 = 1

You can imagine that for 17th century dudes playing at mathematics that could be intimidating to figure out. But they could actually figure it out, which is cool.

But why is that thing as large as it is? At D=60, things are simpler:

x = 31 and y =4.

312 - 60(4)2 = 1.

And at D = 62, again, things are also--small:

x = 63 and y = 8.

632 - 62(8)2 = 1

So why?

Well I figured out the first full explanation for whenever.

My own research proves that D is key (not a big surprise there), and if D is a prime or two times a prime, then solutions will be the largest, with exceptions, when D-1, D+1, D-2, or D+2 is a square, as x is forced to equal 1 or -1 mod D, especially if D-1 has small prime factors, and if 4 is also factor they will be even bigger still.

That is, x-1 or x+1 must have D as a factor, if D is a prime or twice a prime. Which is the biggest rule, and the main thing is x+1 or x-1 having D as a factor, which forces a larger solution unless D-1, D+1, D-2 or D+2 is a square.

And when a larger solution is forced, if D-1 has small prime factors, solutions will be even larger. And then if 4 is also a factor they are the largest.

So now that you know the rules, let's think again about D = 61.

Here notice that x = -1 mod 61, as: 1766319049 + 1 = (28956050)(61)

So that's one thing, where 61 is prime, so not a big surprise, by the rules with x = -1 mod 61. Next, D-1, D+1, D+2, and D-2 are not square, which would override it. That is, if they were square, then the solution would have been forced to be smaller again.

And finally D-1 = 60, which has the first three primes as factors, as 4(3)5) = 60, and 4 is a factor as well, which pushes it way up there.

And that is the 'why' of that solution..

So I can explain why D=61 is so large and my explanation is backed by a mathematical proof which has been public on this blog since September 2011.

Notice the rules--which cover ALL cases--explain the smaller D = 62 solution as being because D+2 is 64, which is a square.

Without explanation though it's still the same equation, of course, known for a long time.

James Harris