On September 7, two mathematicians posted a proof of a version of one of the most famous open problems in mathematics. The result opens a new front in the study of the “twin primes conjecture,” which has bedeviled mathematicians for more than a century and has implications for some of the deepest features of arithmetic.
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
“We’ve been stuck and running out of ideas on the problem for a long time, so it’s automatically exciting when anyone comes up with new insights,” said James Maynard, a mathematician at the University of Oxford.
The twin primes conjecture concerns pairs of prime numbers with a difference of 2. The numbers 5 and 7 are twin primes. So are 17 and 19. The conjecture predicts that there are infinitely many such pairs among the counting numbers, or integers. Mathematicians made a burst of progress on the problem in the last decade, but they remain far from solving it.
The new proof, by Will Sawin of Columbia University and Mark Shusterman of the University of Wisconsin, Madison, solves the twin primes conjecture in a smaller but still salient mathematical world. They prove the conjecture is true in the setting of finite number systems, in which you might only have a handful of numbers to work with.
These number systems are called “finite fields.” Despite their small size, they retain many of the mathematical properties found in the endless integers. Mathematicians try to answer arithmetic questions over finite fields, and then hope to translate the results to the integers.
“The ultimate dream, which is maybe a bit naive, is if you understand the finite field world well enough, this might shed light on the integer world,” Maynard said.
In addition to proving the twin primes conjecture, Sawin and Shusterman have found an even more sweeping result about the behavior of primes in small number systems. They proved exactly how frequently twin primes appear over shorter intervals — a result that establishes tremendously precise control over the phenomenon of twin primes. Mathematicians dream of achieving similar results for the ordinary numbers; they’ll scour the new proof for insights they could apply to primes on the number line.
A New Kind of Prime
The twin primes conjecture’s most famous prediction is that there are infinitely many prime pairs with a difference of 2. But the statement is more general than that. It predicts that there are infinitely many pairs of primes with a difference of 4 (such as 3 and 7) or 14 (293 and 307), or with any gap of 2 or larger that you might want.
Alphonse de Polignac posed the conjecture in its current form in 1849. Mathematicians made little progress on it for the next 160 years. But in 2013 the dam broke, or at least sprung major leaks. That year Yitang Zhang proved that there are infinitely many prime pairs with a gap of no more than 70 million. Over the next year other mathematicians, including Maynard and Terry Tao, closed the prime gap considerably. The current state of the art is a proof that there are infinitely many prime pairs with a difference of at most 246.
But progress on the twin primes conjecture has stalled. Mathematicians understand they’ll need a wholly new idea in order to solve the problem completely. Finite number systems are a good place to look for one.
To construct a finite field, start by extracting a finite subset of numbers from the counting numbers. You could take the first five numbers, for instance (or any prime number’s worth). Rather than visualizing the numbers along a number line the way we usually do, visualize this new number system around the face of a clock.
Arithmetic then proceeds, as you might intuit it, by wrapping around the clock face. What’s 4 + 3 in the finite number system with five elements? Start at 4, count three spaces around the clock face, and you’ll arrive at 2. Subtraction, multiplication and division work similarly.
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Illustration: Lucy Reading-Ikkanda/Quanta Magazine
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