# Caltech mathematicians solve 19th-century number mystery – finally prove “Patterson’s conjecture”

### Caltech mathematicians Alex Dunn and Maxim Radzivil finally prove the “Patterson conjecture.”

A perplexing feature of numbers first encountered by German mathematician Ernst Kummer has baffled researchers for the past 175 years. At one point in the 1950s, this quirky feature of number theory was thought to be wrong, but decades later mathematicians found hints that it was indeed the case. Now, after a few twists and turns, two Caltech mathematicians have finally found proof that Coomer was right all along.

“We had a few ‘aha’ moments, but then we had to roll up our sleeves and figure it out,” explains Alexander (Alex) Dunn, a Caltech doctoral student and mathematics instructor to Olga Tavskyi and John Todd, who wrote the proof with his advisor, professor of mathematics Maksim Radzivil and posted it online in September 2021

The math problem involves Gaussian sums, which are named after the prolific 18th-century mathematician Carl Friedrich Gauss. When Gauss was young, he surprised his classmates quickly develop a formula for adding numbers from 1 to 100. Gauss later developed a complex concept known as Gaussian sums that easily represent the distribution of solutions to equations. He looked at the distribution of so-called Gaussian sums of squares for non-trivial primes (primes that have a remainder of 1 when divided by 3) and found “a beautiful structure,” according to Radziwill.

This addition activity involves a type of math known as modular arithmetic. A simple way to understand modular arithmetic is to think of a clock and its face divided into 12 hours. When it’s noon or midnight, the numbers reset and go back to 1. This “modulo 12” system simplifies timekeeping because we don’t have to count the hours forever.

In the case of Gaussian sums, the same idea applies, but the base “dial” is divided by *old* hours where *old* is a prime number. “Modulo p math is a way to strip away information and make incredibly complex equations simpler,” Radziwill says.

In the 19th century, Coomer was interested in the distribution of cubic Gaussian sums for nontrivial primes, or the system modulo p. He did this by hand for the first 45 non-trivial primes and plotted the answers one by one on the number line (to do this he had to first normalize the answers to be between -1 and 1). The result was unexpected: the solutions were not random, but tended to cluster toward the positive end of the line.

“When we’re dealing with the distribution of natural objects in number theory, the naive expectation is that the distribution is equal, and if it’s not, there has to be a very compelling reason,” says Dunn. “That’s why it was so shocking that Coomer claimed it wasn’t true for cubes.”

Later, in the 1950s, researchers led by the late Hedwig Selberg of the Institute for Advanced Study used a computer to calculate cubic Gaussian sums for all nontrivial primes less than 10,000 (about 500 primes). When the solutions were plotted on a number line, the bias observed by Coomer disappeared. The solutions appeared to have a random distribution.

Then along came the mathematician Samuel Patterson, who in 1978 proposed a solution to the confusion, now known as the Patterson hypothesis. Patterson, who was a graduate student at the University of Cambridge at the time, recognized that bias in the distribution of decisions could become overwhelming as sample sizes got larger and larger. This meant that Coomer was right—something funny was going on with his sums for the 45 primes. But to prove why that is, you have to wait until last year, when Dunn and Radziwill finally figured it out.

“The tilt seen with a few numbers is like having a physically impossible coin that pushes a little bit toward the heads, but it gets smaller and smaller the more times you flip it,” Radziwill explains.

Two Caltech researchers decided to work together to try to solve the problem of Patterson’s hypothesis about two years ago. They didn’t spend much time together on campus because of the pandemic, but they ran into each other in a parking lot in Pasadena and started talking. They decided to meet in the parks to work on a problem where they would write down their mathematical proofs on pieces of paper.

“I had just come to Caltech and I didn’t know a lot of people,” Dunn says. “So it was really nice to meet Max and have the opportunity to work on the problem in person.”

Their solution was based on the work of Roger Heath-Brown of[{” 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: “Displacement in cubic Gaussian sums: Patterson’s conjecture” by Alexander Dunn and Maxim Radzivil, 15 September 2022. *Mathematics > Number theory*.

arXiv:2109.07463

https://scitechdaily.com/caltech-mathematicians-solve-19th-century-number-riddle-finally-prove-pattersons-conjecture/ Caltech mathematicians solve 19th-century number mystery – finally prove “Patterson’s conjecture”