It seems almost like fate that the Einstein Lectures, which alternate between physics, mathematics, and philosophy in honor of their namesake, should return to mathematics this year. 

After all, the world of mathematics is in turmoil. Just two weeks ago, AI company OpenAI released more than 700 AI-generated manuscripts addressing previously unsolved mathematical problems, although they have yet to undergo rigorous mathematical verification. Shortly before that, OpenAI had presented an AI-generated proof concerning a variant of the Navier–Stokes Millennium Prize Problem, which has since been the subject of intense scrutiny and debate. 

Some are calling it nothing less than the most significant moment in the history of mathematics. Seismic shifts appear to be on the horizon, driven by AI—a technology that itself owes its existence to mathematics—and leaving mathematicians anxious about their future. 

It was hardly surprising, then, that the opening of the 2026 Einstein Lectures attracted such considerable interest. The auditorium at the University of Bern was nearly filled to capacity as the audience gathered to hear Canadian-American mathematician Manjul Bhargava. They were rewarded with plenty of magic and a major surprise. 

Mathematics as Magic, Music, and Art 

That Bhargava should be delivering this year's Einstein Lectures seems equally fateful. For one thing, there is his connection to Albert Einstein, who joined the Institute for Advanced Study in Princeton after emigrating to the United States, while Bhargava is now a professor at Princeton University. Bhargava jokingly pointed out, however, that Einstein had made his greatest breakthroughs right here in Bern. 

For another, the mathematician of Indian descent, who was awarded the Fields Medal in 2014 for his work in number theory, is among the most prominent representatives of an intuitive and—in contrast to AI—profoundly human approach to mathematics. 

Throughout his work, Bhargava repeatedly draws on music, poetry, and magic, weaving mathematics into the story of his own life. As a child, he taught himself mathematical theorems through play. As a teenager, he explored the mathematical principles underlying Indian poetry and drumming rhythms. Later, his fascination with a Rubik's Cube led to his first major breakthrough. 

He has also been actively involved in mathematics education, contributing, for example, to India's National Education Policy 2020. Since February of this year, he has served as the inaugural president of the National Museum of Mathematics (MoMath) in New York City. 

This genuine enthusiasm for mathematics—and for sharing it with a wider audience—was palpable on the opening evening of the Einstein Lectures, which began with an appropriately playful question: Bhargava asked the audience to think about their favorite number. 

The Magic Number 142,857 

Bhargava recalled how other children would typically choose a lucky number such as 7 or 11, or perhaps their birthday. His own favorite, however, had always been 142,857. As the audience laughed, he added jokingly that everyone had naturally assumed he was pulling their leg. But as soon became clear, Bhargava had good reasons for his choice. 

The number 142,857 has some astonishing properties. Multiply it by any integer from 1 to 6, and the result contains exactly the same sequence of digits, shifted into a different position: 142,857 × 1 = 142,857; 142,857 × 2 = 285,714; 142,857 × 3 = 428,571; and so on. 

Bhargava explained to his astonished audience that the result was always a cyclic rearrangement of the same digits. Even when multiplied by larger numbers, the same circular pattern emerges, albeit in a somewhat more complicated way. This remarkable property, he explained, makes 142,857 what mathematicians call a cyclic number—and a favorite tool of magicians and sleight-of-hand performers. 

But beneath this entertaining surface lies a connection to some of the deepest structures in mathematics: prime numbers. 

The number 142857 is the repeating block of digits in the decimal expansion of the fraction 1/7, namely 0.142857… The next cyclic number, 0588235294117647, likewise comes from a fraction whose denominator is prime: 1/17, or 0.0588235294117647… Here, the leading zero is part of the cyclic number's digit sequence. 

This led, Bhargava explained, to the question of which prime numbers give rise to cyclic numbers. 

Bhargava succeeded in proving part of the answer last year, establishing a classification of cyclic numbers: They correspond to the repeating blocks in the decimal expansions of fractions of the form 1/p, where p is a prime number for which the repeating block has length p − 1. 

Groundbreaking Results Thanks to AI … 

An even deeper question concerns whether there are infinitely many prime numbers that give rise to cyclic numbers, and how these primes are distributed. This question is addressed by Artin's conjecture, named after Austrian mathematician Emil Artin. As Bhargava explained, however, the conjecture had remained unproven—or rather, it had remained unproven until two weeks earlier. 

Among the more than 700 problems that OpenAI appears to have solved, Bhargava explained, is the so-called quasi-Riemann hypothesis, a weaker form of the famous Riemann hypothesis concerning the distribution of prime numbers—a problem widely regarded as one of the holy grails of mathematics. 

According to Bhargava, mathematicians Cureton and Thorner subsequently derived the generalized quasi-Riemann hypothesis, a weaker form of the generalized Riemann hypothesis. The latter, as it happens, implies Artin's conjecture. 

At this point, one might understandably feel a little disheartened. Surely AI has now solved the mystery of cyclic numbers as well, stripping yet another enchanting corner of mathematics of its magic. 

But Bhargava would not be such a gifted communicator of mathematical ideas if he did not have an unexpected twist in store. 

… and a Surprising New Result 

Bhargava began his big revelation by explaining that his invitation to deliver the Einstein Lectures had motivated him to spend the previous two weeks investigating whether the generalized quasi-Riemann hypothesis, rather than the generalized Riemann hypothesis, could be used to gain new insights into cyclic numbers. 

And indeed, within the preceding week, building on the results of OpenAI, Cureton, and Thorner, Bhargava had succeeded in proving upper and lower bounds for the number of cyclic numbers. These bounds approach the predictions of Artin's conjecture and guarantee that, for a positive proportion of prime numbers p, the fraction 1/p has a decimal expansion with a repeating block of length p − 1. 

With a modest smile, Bhargava concluded that his result established the existence of infinitely many cyclic numbers. 

After a few seconds of absorbing the significance of what they had just heard, the audience erupted into enthusiastic applause. It was quite possibly the first time an entirely new research result had been announced at the Einstein Lectures. 

Facing an Uncertain Future with Confidence 

Reflecting on the experience afterward, Bhargava cheerfully described the discovery as a moment in which his personal journey had come full circle. 

And the evening's story, too, seemed to have come full circle: Once again, a fascination with the beauty and magic of mathematics had helped human intelligence triumph over artificial intelligence. 

Yet the real impact of AI on mathematics cannot be so easily wrapped up in a reassuring story. This became evident during the Q&A session, when several audience members raised the issue. 

Bhargava, however, remained optimistic. He emphasized that humans still had an important role to play in mathematics—not least because AI-generated proofs would be impossible without the foundations laid by human mathematical discoveries. 

Ideally, he suggested, AI could become a tool that mathematicians use in their work and draw inspiration from, just as he himself had done in his research on cyclic numbers. 

One can only hope that this optimism will carry through to the remaining evenings, when Bhargava will lead his audience further into the enchanting world of numbers, exploring magic squares and poetic rhythms.