Fields medalist and number theorist James Maynard, professor of number theory at the Mathematical Institute in Oxford, described the fundamental role of prime numbers at the 2025 Carrier Medal Ceremony and Lecture at the University of Notre Dame in September.
Maynard, who is known for his work on prime gaps and analytic number theory, addressed an audience of nearly 150 people in Jordan Hall of Science about why prime numbers are at once the simplest and most mysterious objects in mathematics—and how they are essential to modern life.
A prime number is a whole number larger than 1 that cannot be written as two smaller whole numbers multiplied together. Therefore, 15 is not prime because it can be divided in whole numbers by 3 and 5. However, 7 is a prime number because it can only be divided by 7 and 1.
“The natural question is, why on earth should you care about prime numbers? This looks like some esoteric thing that maybe math nerds would like, but wouldn't have any relevance,” he joked.
Maynard drew comparisons between primes and the building blocks used in other scientific disciplines. Chemists look at atoms to understand molecules, and geneticists look at DNA base pairs to understand heredity, he said. Similarly, mathematicians look at primes to understand the multiplicative structure of whole numbers.
One reason they are crucial is because they provide insight into how mathematicians think about complex systems, since many whole numbers can be broken down into prime factors in a unique way, a concept formalized in the fundamental theorem of arithmetic, he said.
“If you're trying to understand something that's big and complicated, [the principle] is to break it down into much simpler constituent pieces and then try and build your understanding up from there,” said Maynard, who won the Fields Medal, one of the most prestigious honors in mathematics, in 2022.
One common use for prime numbers involves internet security and online shopping.
“Every day now we're using the internet, and I might try and buy something on Amazon. Maybe I'll try to buy a maths book or something . . . so that's the sort of thing that I buy,” he said, and the crowd laughed. “But when I'm buying my math book, I'm going to give my credit card details, and I want to make sure my credit card details are secure, so a hacker can't steal my credit card details and use them to buy themselves lots of math books.”
The encryption used relies on multiplying large prime numbers together to create an even larger number for the “public key,” which scrambles the credit card number during the encryption process. To decrypt it, however, one must have the two original primes, which are quite difficult to factor out, even by a supercomputer.
After Maynard’s talk, he and Kate Biberdorf, the Professor for Public Understanding of Science, had a fireside chat where audience members could submit questions. Maynard answered questions ranging from how to prepare for a mathematics career beginning as young as middle school, to how to solve parts of important theorems, to how to describe the concept of “sexy primes.” (And no, they might not be what you think: They are prime numbers in a pair or larger group of primes that differ from each other by exactly six.)
The Rev. Joseph Carrier, C.S.C., Science Medal, presented in honor of Mary Burke Ryan, is awarded annually in recognition of sustained, outstanding achievements in the sciences. It celebrates those whose pioneering contributions have advanced the frontiers of scientific knowledge and promoted the betterment of humanity.
Established by the College of Science at the University of Notre Dame in 2022, the medal honors distinguished scientists whose work has transformed their disciplines and whose discoveries inspire future generations. The award rotates annually among the mathematical, physical, chemical, and biological sciences and is accompanied by a monetary prize.
Maynard’s medal was presented by Steve Corcelli, interim William K. Warren Foundation Dean of the College of Science, after introductory remarks by him and David Galvin, professor and chair of the Department of Mathematics.
“The fundamental reason that I think primes are so fascinating is that, on the one hand, primes are amongst the most basic objects that you could possibly think of … but then the big tension is that they remain mysterious,” Maynard said. “Quite quickly, even when you're asking basic questions about primes, you are into famous problems that mathematicians have been suffering to solve for 1000s of years.”
The 2026 Carrier Medal Ceremony and Lecture will be Monday, Sept. 14, with 2018 Nobel Laureate Frances H. Arnold sharing her research about the directed evolution of enzymes, which can be used for biofuels and pharmaceuticals.
Originally published by at carriermedal.nd.edu on September 18, 2025.