Caltech mathematicians Alex Dunn and Maksym Radziwill finally prove 'Patterson's conjecture.' A perplexing feature of numbers first stumbled upon by German mathematician Ernst Kummer has confounded researchers for the past 175 years. At one point in the 1950s, this quirky feature of number theory w
Caltech mathematicians Alex Dunn and Maksym Radziwill finally prove a perplexing feature of numbers first stumbled upon by German mathematician Ernst Kummer. Credit: CaltechA perplexing feature of numbers first stumbled upon by German mathematician Ernst Kummer has confounded researchers for the past 175 years. At one point in the 1950s, this quirky feature of number theory was thought to have been wrong, but then, decades later, mathematicians found hints that it was in fact true.
“When dealing with the distribution of natural objects in number theory, the naive expectation is that one has an equal distribution, and if not, there should be a very convincing reason,” Dunn says. “That is why it was so shocking that Kummer claimed that this wasn’t the case for cubes.” Then came mathematician Samuel Patterson who proposed a solution to the mix-up in 1978, now referred to 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 be overwhelmed as the sample size gets bigger and bigger. That meant Kummer was right—something funny was going on with his sums for 45 primes.
“I had just come to Caltech and didn’t know many people,” Dunn says. “So it was really great to run into Maks and be able to work together on the problem in person.”, 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.
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