# A case study of dogged perseverance in mathematics

It is widely held that mathematics at the highest levels is a young person’s game and that once one hits the age of forty, one has pretty much exhausted one’s potential for any creative contribution to the field. The famous mathematician G. H. Hardy said, “No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man’s game… I do not know of an instance of a major mathematical advance initiated by a man past fifty.”

This was partly why it was such a shock when in 2015 a paper appeared in the prestigious journal Annals of Mathematics claiming to have solved a major unsolved problem and that the author was the nearly 60-year old Yitang Zhang, an untenured part-time calculus teacher at the University of New Hampshire who had published only one paper in the past in 2001.

So what did Zhang do? He proved a long-standing conjecture in number theory involving prime numbers that dates back to the nineteenth century. Alec Wilkinson wrote a long article in the New Yorker about Zhang, the theorem he proved, and how it came about.

No formula predicts the occurrence of primes—they behave as if they appear randomly. Euclid proved, in 300 B.C., that there is an infinite number of primes. If you imagine a line of all the numbers there are, with ordinary numbers in green and prime numbers in red, there are many red numbers at the beginning of the line: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47 are the primes below fifty. There are twenty-five primes between one and a hundred; 168 between one and a thousand; and 78,498 between one and a million. As the primes get larger, they grow scarcer and the distances between them, the gaps, grow wider.

Euclid proved, in one of the most elegant of proofs, that there is no largest prime number, that however far you up in numbers, you will encounter yet another one and another one and so on.

The twin-prime conjecture that Zhang attacked was that:

[N]o matter how far you travel on the number line, even as the gap widens between primes you will always encounter a pair of primes that are separated by two. The twin-prime conjecture is still unsolved. Euclid’s proof established that there will always be primes, but it says nothing about how far apart any two might be.

No one had been able to prove that if you travel a finite distance along the number line, you will always encounter at least two primes within that distance. What Zhang did was show that there is a distance within which there will always be two primes.

Wilkinson explains what his paper means:

Picture it as a ruler that might be applied to the line of green and red numbers. Zhang chose a ruler of a length of seventy million, because a number that large made it easier to prove his conjecture. (If he had been able to prove the twin-prime conjecture, the number for the ruler would have been two.) This ruler can be moved along the line of numbers and enclose two primes an infinite number of times. Something that holds for infinitely many numbers does not necessarily hold for all. For example, an infinite number of numbers are even, but an infinite number of numbers are not even, because they are odd. Similarly, this ruler can also be moved along the line of numbers an infinite number of times and not enclose two primes.

From Zhang’s result, a deduction can be made, which is that there is a number smaller than seventy million which precisely defines a gap separating an infinite number of pairs of primes. You deduce this, a mathematician told me, by means of the pigeonhole principle. You have an infinite number of pigeons, which are pairs of primes, and you have seventy million holes. There is a hole for primes separated by two, by three, and so on. Each pigeon goes in a hole. Eventually, one hole will have an infinite number of pigeons. It isn’t possible to know which one. There may even be many, there may be seventy million, but at least one hole will have an infinite number of pigeons.

Having discovered that there is a gap, Zhang wasn’t interested in finding the smallest number defining the gap. This was work that he regarded as a mere technical problem, a type of manual labor—“ambulance chasing” is what a prominent mathematician called it. Nevertheless, within a week of Zhang’s announcement mathematicians around the world began competing to find the lowest number.

Zhang used a very complicated form of a simple device for finding primes called a sieve, invented by a Greek named Eratosthenes, a contemporary of Archimedes. To use a simple sieve to find the primes less than a thousand, say, you write down all the numbers, then cross out the multiples of two, which can’t be prime, since they are even. Then you cross out the multiples of three, then five, and so on. You have to go only as far as the multiples of thirty-one. Zhang used a different sieve from the one that others had used. The previous sieve excluded numbers once they grew too far apart. With it, Goldston, Pintz, and Yıldırım had proved that there were always two primes separated by something less than the average distance between primes that large. What they couldn’t identify was a precise gap. Zhang succeeded partly by making the sieve less selective.

Interestingly, the fact that Zhang was perfectly content with being an untenured faculty member may have given him the freedom to work on big problems.

Zhang’s preference for undertaking only ambitious problems is rare. The pursuit of tenure requires an academic to publish frequently, which often means refining one’s work within a field, a task that Zhang has no inclination for. He does not appear to be competitive with other mathematicians, or resentful about having been simply a teacher for years while everyone else was a professor. No one who knows him thinks that he is suited to a tenure-track position. “I think what he did was brilliant,” Deane Yang told me. “If you become a good calculus teacher, a school can become very dependent on you. You’re cheap and reliable, and there’s no reason to fire you. After you’ve done that a couple of years, you can do it on autopilot; you have a lot of free time to think, so long as you’re willing to live modestly. There are people who try to work nontenure jobs, of course, but usually they’re nuts and have very dysfunctional personalities and lives, and are unpleasant to deal with, because they feel disrespected. Clearly, Zhang never felt that.”

There is always something fascinating about the lone scholar, working in isolation and obscurity, who suddenly emerges and reveals a major contribution to the field.

1. birgerjohansson says

Going off on a tangent… can we please clone Evariste Galouis (I might be misspelling his name)?
And Srinivasa Ramanujan, if he was buried and not cremated.

Also, Zhang comes across as a sympathic character, sadly not the case with every bona fide genius. 🥀🥂

2. Owlmirror says

G. H. hardy said, ““

Uncapitalized name, and a redundant quotation mark.

(From the quoted article)

There is a hole for primes separated by two, by three,

Maybe it’s pointless to note an error from an article 8 years old, but the third number past one non-even prime (or before one such prime) will always be even, and therefore not prime. Presumably the author meant by four, and then by six, and so on.

[Corrected. Thanks! -MS]

3. Rob Grigjanis says

The “young person’s game” principle also applies to theoretical physics. Of course there are exceptions, but they are rare.