Rough Primes: How Mathematicians Used Fuzziness to Beat Precision

In July 2024, during a tea break at a mathematics conference in Edinburgh, Mehtaab Sawhney met Ben Green, a mathematician he had long admired. Sawhney had just graduated from graduate school; a result Green proved twenty years earlier was, in Sawhney's words, "one of the things that brought me into this field." Green was equally impressed: "Mehtaab is an extraordinary mathematician," he later recalled. "He knows everything."

The two decided to collaborate. They just needed the right problem.

Neither knew that in the following weeks, they would solve the final piece of a puzzle that had stood for 384 years—and that the key tool would come from a field that seemed to have nothing to do with primes.

Fermat's Seed

In 1640, Pierre de Fermat scribbled a conjecture in the margins of a letter: there are infinitely many primes that can be written as the sum of two perfect squares.

13 = 2² + 3². 29 = 2² + 5². 41 = 4² + 5².

A century later, Leonhard Euler proved it. Since then, mathematicians have been playing a game: tighten the constraints, and see whether primes can still survive.

In the 1990s, John Friedlander and Henryk Iwaniec proved there are infinitely many primes of the form a² + b⁴—a square plus a fourth power. In 2018, they posed a trickier question:

> If p and q must themselves be prime, are there infinitely many primes of the form p² + 4q²?

Two examples: 41 = 5² + 4×2² = 25 + 16. And 149 = 7² + 4×5² = 49 + 100. In both cases, 5, 2, 7, 5, and the results 41 and 149 are all prime.

It looks simple. But Green summed up the difficulty:

"The more you constrain a set, the harder it is to find primes in it."

Why It's Hard

Imagine searching for pearls on a beach. Primes are the pearls—sparsely scattered along the number line with no obvious pattern.

Fermat asked: are there infinitely many pearls on the beach? Euler said yes.

The 2018 Friedlander–Iwaniec conjecture asks: what if each pearl must be composed of two smaller pearls? The form p² + 4q² requires p and q themselves to be prime—pearls nested inside pearls.

Traditional counting tools fail here. Mathematicians usually count primes with "sieves"—sifting out non-primes like sand through a mesh. But when the constraints are this tight, the sieve's holes must be so fine that the sieve itself becomes useless.

Green and Sawhney spent days at a blackboard during a week in Oxford. The traditional path was a dead end.

Then they had a strange idea.

Rough Primes: Fuzzy Pearls

If you can't count true primes, can you count numbers that are "almost prime"?

They used a concept called rough primes. The definition is simple: numbers not divisible by the first few small primes (2, 3, 5, 7).

Between 1 and 200, there are 50 rough primes. Of these, 46 are true primes, and 4 are false positives: 121 (= 11²), 143 (= 11×13), 169 (= 13²), and 187 (= 11×17).

Rough primes are slightly more numerous than true primes, but they are far better behaved—easier for mathematicians to understand.

"Rough primes are a set that we understand much, much better," Sawhney said.

Green and Sawhney first proved that there are infinitely many primes expressible as the sum of the squares of two rough primes. That step was relatively easy.

But a question remained: does this imply the version for true primes?

Gowers Norms: A Forgotten Ruler

Rough primes and true primes are different sets. Proving a result for one doesn't automatically yield it for the other. Green and Sawhney needed to show that, for their problem, the two were "equivalent."

This required analyzing functions called Type I and Type II sums. If rough primes and true primes produce the same Type I/II sums, then substituting one for the other is legitimate.

But how could they prove the two sets are "the same from a certain point of view"?

The answer came from an unexpected place.

In the early 2000s, Timothy Gowers of Cambridge (later a Fields medalist) developed a tool called the Gowers norm. It measures how "random" or how "structured" a sequence is.

Gowers norms seem to have nothing to do with counting primes. Sawhney himself said: "As an outsider, it's almost impossible to see how these things are connected."

But in 2018, Terence Tao and Tamar Ziegler proved a landmark result linking Gowers norms to Type I/II sums.

Even more remarkably, earlier that same year, Sawhney had independently developed a technique using Gowers norms to compare sets—for a completely unrelated problem.

All the pieces were there. Green and Sawhney assembled them.

Using Gowers norms, they proved that rough primes and true primes have the same Type I/II sums. The rough-prime result could be "translated" back into the true-prime version.

The Friedlander–Iwaniec conjecture was proved: there are infinitely many primes of the form p² + 4q², with p and q both prime.

Seeing Through Blur

The most fascinating part of this story is not the result, but the method.

Mathematics has a reputation for precision—you either prove something or you don't. Yet Green and Sawhney's path was the opposite: they abandoned precision in order to see the truth.

True primes are slippery. Their distribution is like stars in the night sky—you can see each one, but no pattern. Rough primes are the out-of-focus version—the stars become blurry blobs, but the constellations come into view.

This isn't the first time "fuzziness" has beaten "precision." Physics has a similar concept called coarse-graining—you deliberately give up microscopic detail to see macroscopic laws. A drop of water contains 10²³ molecules; you can't track them all, but fluid dynamics doesn't require it.

Gowers norms are themselves a "blurring" tool. They don't tell you what a sequence looks like in detail—only how random it is. But that single measure of randomness was enough to decide whether two sets are equivalent for counting purposes.

Tamar Ziegler, asked about this result, offered a moving observation:

"It's like being a parent—when you let your children grow freely, they do mysterious and unexpected things."

Her "child" was the Gowers norm. The 2018 result she proved with Tao connected Gowers norms to number theory. Six years later, that connection bore fruit she never anticipated.

The Power of Relaxation

This story suggests a more general lesson.

In AI and engineering, we often face the same dilemma: exact solutions are too expensive—can we use approximations instead?

Linear programming relaxations turn integer programs into continuous problems, and sometimes still yield optimal solutions. Surrogate models approximate complex black-box functions and shine in Bayesian optimization. Probing doesn't tell you what each neuron does—only what information a layer "encodes"—yet that coarse-grained view is enough to understand a model's behavior.

Green and Sawhney's proof shows this isn't laziness—it's strategy.

When you face a problem too precise to even begin, try making it a little fuzzier. In a blurry world, the path often comes into focus.

Fermat planted a seed in 1640. 384 years later, two mathematicians—with a forgotten ruler and some "almost prime" numbers—finally saw it bloom.

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References:

• 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.