The proof of a long-standing conjecture in number theory, concerning the infinite presence of primes expressible in the form p² + 4q², where p and q are also primes, has been achieved by mathematicians Ben Green and Ashwin Sahwney. This breakthrough relies heavily on an unexpected mathematical tool: the Gowers norm, a concept traditionally used to measure the randomness or structure within sets of numbers and functions. The journey to this proof involved navigating a complex landscape of mathematical constructs, connecting seemingly disparate concepts in a novel and insightful manner.
Green and Sawhney’s initial challenge was to prove the existence of infinitely many primes within specific families. Their approach hinged on analyzing specialized functions, known as Type I and Type II sums, associated with different versions of their problem. The core objective was to demonstrate the equivalence of these sums, regardless of the constraints applied. This equivalence would validate the use of “rough” primes, approximations of actual primes, within their proof without sacrificing the integrity of their calculations. This substitution was crucial in simplifying the problem to a manageable level.
The mathematicians realized that the key to demonstrating this equivalence lay within the Gowers norm, a tool each had encountered independently in their previous research. The Gowers norm, developed by mathematician Timothy Gowers, quantifies the randomness or structuredness of a function or set of numbers. At first glance, the relevance of the Gowers norm to this particular number theory problem seemed tenuous, almost belonging to a different mathematical universe. However, Green and Sawhney, recognizing the potential, sought a way to bridge this gap.
Their bridge came in the form of a 2018 result by mathematicians Terence Tao and Tamar Ziegler. This result allowed Green and Sawhney to establish a connection between Gowers norms and the crucial Type I and Type II sums. Specifically, they needed to use the Gowers norm to demonstrate a sufficient similarity between two sets of primes: one set constructed using the approximate “rough” primes and the other using actual primes. The similarity, in terms of their Gowers norms, would imply the desired equivalence of their Type I and II sums.
Serendipitously, Sawhney had previously developed a technique for comparing sets using Gowers norms while tackling an entirely separate problem. Remarkably, this technique proved to be precisely what was needed to establish the equivalence of the Type I and II sums for the two sets of primes. This equivalence, established via the Gowers norm, served as the crucial linchpin in Green and Sawhney’s proof.
With this hurdle overcome, Green and Sawhney successfully proved Friedlander and Iwaniec’s conjecture, confirming the infinite existence of primes of the form p² + 4q². Their success extended beyond this specific form, demonstrating the infinite presence of primes within other families as well. This achievement represents a significant advancement in a field where breakthroughs are infrequent and hard-won. The implications extend beyond the immediate problem, showcasing the Gowers norm as a potent tool in this area of number theory.
Beyond the specific proof, the work heralds a potentially transformative introduction of the Gowers norm into number theory. The novelty of its application in this context opens up exciting avenues for future research. Mathematicians are now exploring the potential of the Gowers norm to tackle other long-standing problems in number theory, extending its reach beyond the counting of primes. This innovative use of the Gowers norm signifies a paradigm shift, enriching the toolset available to number theorists and promising further advancements in the field. The unexpected connection between seemingly disparate areas of mathematics underscores the interconnected nature of the discipline and the potential for novel applications of existing tools.
