Caltech Mathematicians Prove Patterson

Caltech Mathematicians Solve 19th-Century Number Puzzles – Finally Prove “Patterson’s Conjecture”

Caltech mathematicians Alex Dunn and Maksym Radziwill finally prove a puzzling feature of numbers first encountered by German mathematician Ernst Kummer. Photo credit: Caltech

Finally, Caltech mathematicians Alex Dunn and Maksym Radzi will prove “Patterson’s Conjecture.”

A baffling feature of numbers, first discovered by German mathematician Ernst Kummer, has baffled researchers for the past 175 years. At one point in the 1950s, this quirky property of number theory was thought to be false, but then, decades later, mathematicians found evidence that it was in fact true. Now, after several twists and turns, two Caltech mathematicians have finally found proof that Kummer was right all along.

“We had several ‘aha’ moments, but then you have to roll up your sleeves and find out,” explains Alexander (Alex) Dunn, a postdoc at Caltech and an Olga Taussky and John Todd Instructor in Mathematics, who wrote the proof with his advisor , mathematics professor Maksym Radziwill, and put it online in September 2021.

The math problem has to do with Gaussian sums, named after the prolific 18th-century mathematician Carl Friedrich Gauss. When Gauss was young, he amazed his classmates by quickly devising a formula for adding the numbers 1 through 100. Gauss later developed a complex concept known as Gauss sums, which easily depicts the distribution of solutions to equations. He examined the distribution of so-called square Gaussian sums for non-trivial prime numbers (prime numbers that have a remainder of 1 when divided by 3) and, according to Radziwill, found a “nice structure”.

Maxim Radziwill

Maksym Radziwill, Professor of Mathematics. Photo credit: Caltech

This summation activity involves a type of mathematics known as modular arithmetic. A simple way to understand modular arithmetic is to imagine a clock whose face is divided into 12 hours. When it comes to noon or midnight, the numbers reset and go back to 1. This “modulo 12” system simplifies timekeeping because we don’t have to keep counting up hours.

In the case of Gaussian sums, the same idea is at play, but the base “dial” is divided into p hours where p is a prime number. “Modulo-p math is a way of filtering out information and making incredibly complicated equations simpler,” says Radziwill.

In the 19th century, Kummer was interested in studying the distribution of cubic Gaussian sums for nontrivial primes or in a modulo p system. He did this by hand for the first 45 non-trivial primes, plotting the answers individually on a number line (to do this he first had to normalize the answers so that they fell between -1 and 1). The result was unexpected: the solutions were not random but tended to cluster towards the positive end of the line.

“In number theory, when you’re looking at the distribution of natural objects, the naïve expectation is that you have a uniform distribution, and if you don’t, there should be a very compelling reason,” says Dunn. “That’s why it was so shocking that Kummer claimed this wasn’t the case with dice.”

Alex Dun

Alex Dunn, Postdoctoral Researcher and Olga Taussky and John Todd Lecturer in Mathematics. Photo credit: Caltech

Later, in the 1950s, researchers led by the late Hedvig Selberg of the Institute for Advanced Study used a computer to calculate the cubic Gaussian sums for all non-trivial primes below 10,000 (about 500 primes). When the solutions were plotted on the number line, the distortion observed by Kummer disappeared. The solutions appeared to be randomly distributed.

Then came the mathematician Samuel Patterson, who in 1978 proposed a solution to the confusion that is now known as Patterson’s conjecture. Patterson, who was a graduate student at the University of Cambridge at the time, recognized that the bias in the distribution of the solutions could become overwhelming as the sample size continues to increase. That meant Kummer was right – something funny was going on with his calculations for 45 primes. But proving why that’s the case would have to wait until last year, when Dunn and Radziwill finally figured it out.

“The distortion you see on a couple of numbers is like having a physically impossible coin that’s weighted slightly towards heads, but diminishes the more you flip it,” explains Radziwill.

The two Caltech researchers decided to work together about two years ago to try to solve the problem of Patterson’s conjecture. They hadn’t spent much time together on campus because of the pandemic, but they met in a Pasadena parking lot and struck up a conversation. They decided to meet in Parks to work on the problem, where they wrote down their mathematical proofs on sheets of paper.

“I had just joined Caltech and didn’t know many people,” says Dunn. “So it was really great to meet Maks and work on the issue personally.”

Their solution was based on the work of Roger Heath-Brown of the

University of Oxford
The University of Oxford is a collegiate research university in Oxford, England, made up of 39 constituent colleges and a number of academic departments organized into four departments. Founded around 1096, it is the oldest university in the English-speaking world and the second oldest continuously operating university in the world after the University of Bologna.

” data-gt-translate-attributes=”[{” attribute=””>University of Oxford, who had seen a talk by Patterson at the University of Cambridge in the late 1970s. Heath-Brown and Patterson teamed up to work on the problem, and then, in 2000, Heath-Brown developed a tool known as a cubic large sieve to help prove Patterson’s conjecture. He got close but the complete solution remained out of reach.

Dunn and Radziwill cracked the problem when they realized that the sieve wasn’t working properly, or had a “barrier” that they were able to remove.

“We were able to recalibrate our approach. In math, you can get trapped into a certain line of thinking, and we were able to escape this,” Dunn says. “I remember when I had one of the ‘aha’ moments, I was so excited that I ran to find Maks at the Red Door [a café at Caltech] and asked him to come to my office. Then we started the hard work of figuring it all out.”

Reference: “Bias incubic Gauss sums: Patterson’s conjecture” by Alexander Dunn and Maksym Radziwill, September 15, 2022, Mathematics > Number Theory.

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