Rough Primes: How Approximation Helped Mathematicians Count Primes
In July 2024, at a mathematics conference in Edinburgh, Mehtaab Sawhney met Ben Green, a mathematician whose work had inspired Sawhney to enter the field. Soon afterward, they began working together on a problem whose solution would unexpectedly involve a concept that appears to have little to do with prime numbers.
Their work resolved the final part of a 384-year-old mathematical puzzle by using “rough primes” and Gowers norms.
1. The Seed Planted by Fermat
In 1640, Pierre de Fermat conjectured that infinitely many primes can be expressed as the sum of two perfect squares. Examples include:
- 13 = 2² + 3²
- 29 = 2² + 5²
- 41 = 4² + 5²
- 41 = 5² + 4×2² = 25 + 16
- 149 = 7² + 4×5² = 49 + 100
- 121 = 11²
- 143 = 11×13
- 169 = 13²
- 187 = 11×17
- Green, B., & Sawhney, M. (2024). *Primes of the form p² + nq²*. arXiv:2410.04189
- Howlett, J. (2024). “Mathematicians Uncover a New Way to Count Prime Numbers.” Quanta Magazine, December 11.
- Columbia News (2025). “A Math Professor Has a New Finding on Primes.” April 22.
- Friedlander, J., & Iwaniec, H. (2018). Conjecture on primes of the form p² + 4q².
Leonhard Euler proved the result a century later. Mathematicians then began strengthening the restrictions, asking how far the set of primes could be constrained while still remaining infinite.
In the 1990s, John Friedlander and Henryk Iwaniec proved that infinitely many primes have the form a² + b⁴, involving one square and one fourth power. In 2018, they proposed a more difficult question: Are there infinitely many primes of the form p² + 4q², where p and q are themselves prime?
For example:
In each case, the smaller numbers and the resulting number are all prime.
As Green explained, the difficulty is straightforward: the more tightly a set is constrained, the harder it becomes to find primes within it.
2. Why the Problem Was Difficult
Primes are sparse among the integers, without an obvious distribution pattern. Fermat’s original question asked whether infinitely many such prime “pearls” exist. Euler established that they do.
The later conjecture asked whether a prime must also be constructed from two smaller primes. The form p² + 4q² therefore creates a nested prime condition: the number must be prime, while p and q must also be prime.
Mathematicians commonly use sieve methods to count primes, analogous to filtering non-primes from sand. With such strong restrictions, however, the sieve must become so fine that it becomes ineffective.
Green and Sawhney spent several days discussing the problem at Oxford without finding a workable route through the traditional method.
3. Rough Primes: Blurry Pearls
The researchers considered a different question: if exact primes were too difficult to count directly, could they count numbers that were “almost prime”?
They used a concept called rough primes. A rough prime is an integer not divisible by the first few small primes, such as 2, 3, 5, and 7.
Between 1 and 200, there are 50 rough primes. Of these, 46 are genuine primes and four are false positives:
Rough primes are slightly more numerous than ordinary primes, but their distribution is more regular and easier to analyze.
As Sawhney explained, “Rough primes are a set that we understand much better.”
Green and Sawhney first proved that infinitely many primes can be written as the sum of the squares of two rough primes. This was the easier step. The difficult question was whether this result could imply the corresponding result for genuine primes.
4. Gowers Norms: A Forgotten Measuring Tool
Rough primes and genuine primes are different sets, so a proof about rough primes does not automatically transfer to ordinary primes. Green and Sawhney needed to show that the two sets were equivalent in a way relevant to their counting problem.
The connection involved functions associated with Type I and Type II sums. If rough and genuine primes produce the same relevant Type I and Type II sums, the replacement could be justified.
The required bridge came from Gowers norms, tools developed in the early 2000s by Timothy Gowers. Gowers norms measure how random or structured a sequence is.
At first glance, this appears unrelated to prime counting. Sawhney noted that, to an outsider, it is almost impossible to see how these subjects are connected.
The crucial link came from a 2018 result by Terence Tao and Tamar Ziegler connecting Gowers norms with Type I and Type II sums. Earlier in 2018, Sawhney had independently developed a technique for comparing sets using Gowers norms while working on a completely different problem.
Green and Sawhney combined these ideas. They showed that rough primes and genuine primes have the same relevant Type I and Type II sums. This allowed them to translate the rough-prime result back into a statement about ordinary primes.
They proved that infinitely many primes have the form p² + 4q², where p and q are both prime. This resolved the Friedlander–Iwaniec conjecture.
5. Blurring Can Reveal Structure
The most striking aspect of the proof is its method. Mathematics is often described as a discipline of exactness: a statement is either proved or not proved. Green and Sawhney’s approach moved in the opposite direction. By relaxing exactness, they made the underlying structure visible.
Genuine primes are difficult to characterize and distribute irregularly. Rough primes provide a blurred version of that distribution. Like turning individual stars into patches of light, the approximation can reveal a pattern that exact data conceals.
This resembles coarse-graining in physics: microscopic details are deliberately ignored to expose large-scale behavior. A volume of water may contain 10²³ molecules, but fluid dynamics does not require tracking every molecule.
Gowers norms perform a similar function. They do not describe every detail of a sequence, but they measure its randomness sufficiently well to determine whether two sets are equivalent for counting purposes.
Tamar Ziegler described the broader idea in memorable terms: it is like being a parent whose children, when given freedom, do mysterious and unexpected things. The “children” in this case were Gowers norms. Her 2018 collaboration with Tao connected Gowers norms to number theory; six years later, that connection produced consequences they had not anticipated.
6. The Power of Relaxation
The proof suggests a general lesson for mathematics, AI, and engineering: when an exact problem is too expensive or difficult to solve directly, a carefully chosen approximation may be the right strategy.
Linear-programming relaxations turn integer problems into continuous ones and can still produce optimal solutions. Surrogate models approximate complex black-box functions and are widely used in Bayesian optimization. Representation probing does not explain what every individual neuron does, but it can reveal what information a layer encodes.
These methods do not give up rigor. They discard details that are unnecessary for the question at hand.
Green and Sawhney’s proof shows that approximation is not necessarily a shortcut or a form of laziness. It can be a deliberate way to expose structure. When a problem is too exact to approach directly, making it slightly rougher may reveal the path toward a rigorous solution.
Fermat planted the original seed in 1640. More than 384 years later, two mathematicians used rough primes and Gowers norms to show that the constrained family of primes is still infinite.