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Ancient Equations Offer New Look at Number Groups - Quanta Magazine

Ancient Equations Offer New Look at Number Groups - Quanta Magazine

Ancient Equations Offer New Look at Number Groups - Quanta Magazine
Aug 10, 2022 3 mins, 33 secs

How often are there integer solutions for equations of the form x2 – dy2 = –1.

Here, d is an integer — a positive or negative counting number — and Archimedes was looking for solutions where both x and y are integers as well.

This class of equations, called the Pell equations, has fascinated mathematicians over the millennia since.

Some centuries after Archimedes, the Indian mathematician Brahmagupta, and later the mathematician Bhāskara II, provided algorithms to find integer solutions to these equations.

In the mid-1600s, the French mathematician Pierre de Fermat (who was unaware of that work) rediscovered that in some cases, even when d was assigned a relatively small value, the smallest possible integer solutions for x and y could be massive.

(As for Archimedes, his riddle essentially asked for integer solutions to the equation x2 – 4,729,494y2 = 1. “To print out the smallest solution, it takes 50 pages,” said Peter Koymans, a mathematician at the University of Michigan. “In some sense, it is a gigantic troll by Archimedes.”).

But the solutions to the Pell equations can do much more.

Now Koymans and Carlo Pagano, a mathematician at Concordia University in Montreal, have proved a decades-old conjecture related to the Pell equation, one that quantifies how often a certain form of the equation has integer solutions.

In the early 1990s, Peter Stevenhagen, a mathematician at Leiden University in the Netherlands, was inspired by some of the connections he saw between the Pell equations and group theory to make a conjecture about how often these equations have integer solutions.

Intriguingly, a class group’s typical behavior is inextricably intertwined with the behavior of Pell equations.

The new work involves the negative Pell equation, where x2 – dy2 is set to equal −1 instead of 1.

In contrast to the original Pell equation, which always has an infinite number of integer solutions for any d, not all values of d in the negative Pell equation yield an equation that can be solved.

In fact, there are a lot of values of d for which the negative Pell equation can’t be solved: Based on known rules about how certain numbers relate to one another, d cannot be a multiple of 3, 7, 11, 15 and so on.

But even when you avoid those values of d and consider only the remaining negative Pell equations, it’s still not always possible to find solutions.

Of the values for d that might work (that is, values that are not multiples of 3, 7, etc.), he predicted that approximately 58% would give rise to negative Pell equations with integer solutions.

Stevenhagen’s guess was motivated in particular by the link between the negative Pell equation and the Cohen-Lenstra heuristics on class groups — a link that Koymans and Pagano exploited when, 30 years later, they finally proved him correct.

In that work, which was published in the Annals of Mathematics, the mathematicians Étienne Fouvry and Jürgen Klüners showed that the proportion of values of d that would work for the negative Pell equation fell within a certain range.

Not only was it the first major step in cementing those broader conjectures as mathematical fact, but it involved precisely the piece of the class group that Koymans and Pagano needed to understand in their work on Stevenhagen’s conjecture.

(If that had been possible, Smith himself would probably have done so.) Smith’s proof was about class groups associated to the right number rings (ones in which $latex \sqrt{d}$ gets adjoined to the integers) — but he considered all integer values of d.

And when Koymans and Pagano tried to apply his techniques to just the class groups they cared about, the methods broke down immediately.

Moreover, they weren’t just characterizing one class group, but rather the discrepancy that might exist between two different class groups (doing so would be a major part of their proof of Stevenhagen’s conjecture) — which would also require some different tools.

(In their paper, Koymans and Pagano list about a dozen conjectures they hope to use their methods on. Many have nothing to do with the negative Pell equation or even class groups.)

But many of the same roadblocks that Koymans and Pagano had to confront are also present in these other contexts

The new work on the negative Pell equation has helped dismantle these roadblocks

Summarized by 365NEWSX ROBOTS

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