Poulet’s Propeller: Musings on Math and Mathculinity

The Penguin Dictionary of Curious and Interesting Numbers (1986) is one of my favourite books. It’s a fascinating mixture of math, mathecdote and math-joke:

2·618 0333…

The square of φ, the golden ratio, and the only positive number such that √n = n-1. (pg. 45)


Kepler discussed the 6-fold symmetry of snowflakes, and attempted to explain it by considering the close packing of spheres in a hexagonal array. (pg. 69)


This appears to be the first uninteresting number, which of course makes it an especially interesting number, because it is the smallest number to have the property of being uninteresting.

It is therefore also the first number to be simultaneously interesting and uninteresting. (pg. 120)

David Wells, who wrote the Dictionary, “had the rare distinction of being a Cambridge scholar in mathematics and failing his degree”. He must be the mathematical equivalent of the astronomer Patrick Moore: a popularizer responsible for opening many minds and inspiring many careers. He’s also written books on geometry and mathematical puzzles. But not everyone appreciates his efforts. This is a sideswipe in a review of William Hartston’s The Book of Numbers:

Thankfully, this book is more concerned with facts than mathematics. Anyone wanting to learn more about [π] or the Fibonacci sequence should turn to the Penguin Dictionary of Curious and Interesting Numbers, a volume which none but propeller-heads will find either curious or interesting. (Review in The Independent, 18th December 1997)

I must be a propeller-head then. But why? I don’t know. I didn’t like maths when I was young and I’ve never been very good at it. Nowadays, however, my motto is Mathematica Magistra Mundi, “Mathematics is the Mistress of the World”, and I see maths everywhere. Or rather, I see that maths is everything, including me and my act of seeing. But the human activity of maths remains a good way of escaping the messy and contingent everyday world, where chance seems to rule, not necessity. But chance is a mathematical phenomenon too and necessity can manifest itself in apparently random ways. It’s necessary that primes, or numbers divisible by only themselves and 1, go on for ever. But it wasn’t necessary (in the same sense) that this should have been proved by the ancient Greeks. Or by anyone at all.

Human beings existed for many thousands of years before maths was invented and maths might have remained a purely practical subject. The infinitude of the primes or the true nature of √2, the square root of two, is irrelevant to things like agriculture, architecture and commerce. So what explains that very strange and subtle book, Euclid’s Elements from 300 B.C.? How did men — and it is overwhelmingly men — evolve to care about mathematical abstraction? What motivated them to discover and analyse entities, like √2 and π, that are found nowhere in the internal or external universe? More fundamentally, what enabled them to do so?

Those are some Big Questions raised by, but not in, The Penguin Dictionary of Curious and Interesting Numbers. It’s a book devoted entirely to numbers: no food, no relationships, no sex, drugs and rock’n’roll. Those things are all mathematical too, but the maths is hidden, implicit, obscured. In Wells’ Dictionary the maths is out in the open: dry, dusty and absolutely delicious.

Delicious to me and many others, that is. David Wells opens the book with “A List of Mathematicians in Chronological Sequence”. The list starts with the Egyptian Ahmes (c. 1650 B.C.), carries on through giants like Archimedes (c. 287-212 B.C.), Euler (1707-83) and Gauss (1777-1885), and ends with the Indian Ramanujan Srinivasa (1887-1920). Ramanujan stands with Évariste Galois (1811-32) as one of the most romantic and tragic figures in mathematical history. He did great things in his few years. But my interest is piqued by someone who is in the book but isn’t in the list: Paul Poulet (1887-1946), a self-taught mathematician from Belgium. Poulet isn’t in the list because he wasn’t a great mathematician. But he was a great mathematizer, as Wells describes in this entry of his Dictionary:


The start of a remarkable sociable chain of no fewer than 28 numbers discovered by Poulet in 1918. [Beiler] Starting at the top of the left column and leading down, the sum of the proper divisors of each number is equal to the next number, 17716 finally leading back to 14316:

 14316  629072  275444  97946
 19116  589786  243760  48976
 31704  294896  376736  45946
 47616  358336  381028  22976
 83328  418904  285778  22744
177792  366556  152990  19916
295488  274924  122410  17716

No other sociable chain is known as large, or larger than, this one, despite its venerable age. (pg. 174)

That final comment was written in 1986. It’s still true in 2013: the chain is even more venerable and still unsurpassed. The largest prime mentioned in Wells’ Dictionary, 2^216091-1, has a piddling 65,050 digits. The current record has 17 million digits. But Poulet’s 28-link sociable chain is still by far the largest known, though he worked with only pen, paper and his own brain. Modern chain-hunters use computers and have tracked down chains with huge numbers like these:

24604969601522730119812483081690177911114675451920256511530819984744010 → 29380170319863500892339892502331399034100294620489917702058830906007990 → 35082116418090090751430321709203100961767254346549696524874398302465610 → 29380170319863500892339892502360608774950013665995101325117337838539190 → 24604969601522730119812483081690177911114675451920256511530819984744010 (See David Moews’ Perfect, amicable and sociable numbers)

Poulet could have jumped to the moon as easily as factorize numbers of that size. Factorizing, or finding the primes that evenly divide a number, is simple to define but can be very difficult to do. To find his 28-link sociable chain — and another chain of five links that starts with 12496 — Poulet had to investigate thousands of numbers and perform thousands of calculations. The prime factors of 14316 are 2^2, 3 and 1193, so its proper divisors are 1, 2, 3, 4, 6, 12, 1193, 2386, 3579, 4772 and 7158. These sum to 19116, whose prime factors are 2^2, 3^4 and 59. The proper divisors created from those sum to 31704, whose prime factors are 2^3, 3 and 1321. And so on. A modern computer can start checking 1, 2, 3… and find that chain in milliseconds. Throw in programming time and you’re talking minutes.

But Poulet must have worked for years and devoted thousands of hours to finding the 12496-chain and 14316-chain. In 1918 he published a paper announcing his discovery. It ends with this laconic comment: Il est inutile, je pense, d’essayer les nombres inférieurs à 12000 que j’ai tous examinés — “It will be pointless, I think, to try numbers below 12000, because I have tested all of them.” That year of publication, 1918, is important and adds to the interest of Poulet’s work. Did he fight during the First World War? Did he hear big guns booming as he calculated prime factors and added proper divisors? It’s an odd thing to think that, as the First World War ground its way through millions of corpses, a Belgian mathematician was quietly hunting down a remarkable sequence of numbers.

Not that he knew it was there: amicable numbers like 220 and 284, whose proper divisors form mutual sums, had been known for centuries, but sociable chains went unsuspected. And their discovery might easily have had to wait until the computer age. A less hard-working, or less obsessive, man than Poulet might have given up long before he reached 12496 and discovered that its proper divisors sum to 14288, whose proper divisors sum to 15472, whose proper divisors sum to 14536, whose proper divisors sum to 14264, whose proper divisors sum to 12496. Voilà! Sociable numbers!

That is a remarkable discovery, but nothing like what awaited Poulet at 14316. The sequence beginning there could have ended in a prime number, whose only proper divisor is 1. Or it could have climbed beyond Poulet’s powers of calculation, like the sequence seeded by the innocent-seeming 138. But the 14316-sequence didn’t do either of those things. It climbed at first, reaching a peak of 629072 = 2^4 x 39317, then fell slowly back, back, back to… 14316. How did Poulet feel when he realized he’d found such a long sequence? It was a remarkable award for a remarkable amount of work. Its practical importance was nil, but in a sense it was far more important than the war amid which it was discovered. Military conflict and mass slaughter are contingent phenomena. They may or may not happen. But the 14316-chain is necessary, something that must be so in the eternal world of arithmetic. If we had to choose an epitaph for mankind and had to make the epitaph as simple as possible, 14316 might be a good choice. Or rather, 11011111101100 might be. That’s 14316 in binary, the simplest of mathematical codes and the likeliest to be understood by aliens.

Aliens who came across “11011111101100” and deduced what it meant would also deduce that there were high intelligence and advanced civilization behind it. They might think it was a mechanical discovery too, something overseen by an intelligent being but actually found by a machine. But it wasn’t. Not here on earth. Elsewhere in the universe, the 14316-chain may have been found the easy way, with a calculating machine. On earth, it was discovered the hard way, by hand. The mathematics behind the concept of sociable chains isn’t hard to understand and the necessary techniques had been around for millennia. But billions of human beings were born and died before Paul Poulet appeared with the necessary obsession, application and leisure to hunt down 12496 and 14316. “Hunt” is the mot juste in more ways than one: in 1929, Poulet published a book called La chasse aux nombres, or The Hunt for Numbers. Unlike mathematics, hunting has always been practised by humans. It may have been a central part of our evolution and perhaps it helped forge the mathematical brain.

Humans hunt by sight, after all, not scent. And we don’t need to see the animals we’re hunting: we can track them by the signs they leave: prints, disturbed vegetation, dung and so on. A trail is linear, like a piece of text or mathematics. Language came before mathematics and was surely essential in its development. Written language in particular. But is the ability to follow and decipher an animal’s trail related to the ability to follow and decipher a string of words or string of mathematical symbols? Perhaps it is. But hunting is more obviously related to mathematics through masculinity: both hunting and maths are heavily dominated by men. It was unlikely that the Poulet who discovered sociable chains could ever have been born Pauline rather than Paul. Very unlikely: the far right of the mathematical bell curve is occupied more and more strongly by men. So Paul Poulet was a propeller-head with two propellers: mathematics and masculinity. He hunted numbers in the way his remote ancestors hunted mammoth or bison. And perhaps he was using adapted tracking modules in the brain as he performed his calculations. He wasn’t a great mathematician, but he remains a thought-provoking one. And he has a secure niche in mathematical history. Si monumentum requiris: 14316. If you need a monument for him: 14316.

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