Heegner Nos.: rooted in pi, with or without e

  • Who would have thought that the Heegner numbers would take in names from Gauss to Gardner, Chudnovsky to Ramanujan, Euler to Patnaik?
  • There are, I’ll have you know, exactly nine Heegner numbers. Not less, not more. Those nine Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67 and 163

It’s April, which means I will arbitrarily assume it’s time to discuss Heegner numbers. There are, I’ll have you know, exactly nine Heegner numbers. Not less, not more. This was a conjecture first made by the great 19th century German mathematician Carl Friedrich Gauss, though he didn’t know them as Heegner numbers. That label came a century later, in 1952, when another German, Kurt Heegner, proved his conjecture.

Those nine Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67 and 163.

But what are they and what do we do with them? Why have I drawn your attention to Heegners, and am I up to the challenge of writing a whole column around them? Well, I’m going to try. For even if you’ve not heard of Kurt Heegner, the story involves several other remarkable figures in mathematics and science, and possibly even beyond, Gauss being just one.

The prolific 18th century mathematician Leonhard Euler gave us this most interesting formula:

• n2 - n + 41

Why is it interesting? Take a look: Set n to 1, and the formula generates 41. Set it to 2, you get 43. Three generates 47. Four gives us 53…and perhaps you’re seeing a pattern here. 41, 43, 47 and 53 are all prime numbers. Had Euler hit upon a holy grail of mathematics, a formula to generate primes?

Now, we know there is no such formula. But starting with 1 and counting up, Euler’s prime-generating polynomial gives us a prime for every value of n, until 41. (n as 41 gives us 412, or 1681, obviously not a prime). So Euler had a holy grail, but it worked only up to a point. But even so, this was heady stuff. It got other mathematicians excited: Are there other such formulae?

Yes, as it turns out. In 1913, Russian physicist Georg Rabinovitch showed there are other prime-generating formulae that look similar to Euler’s, but only if they satisfy this arcane condition: If you take that last number in the formula (41, here), multiply it by 4 and subtract 1, you must get a Heegner number. Try it: 41 x 4 - 1 is 163, the largest Heegner number— and that’s why Euler’s formula generates primes. The others of this form that work in the same way are those linked to the Heegner numbers 7, 11, 19, 43, 67 and 163; that is, prime-generating formulae—go ahead, work them out—that end in 2, 3, 5, 11, 17 and 41.


A French engineer, François Le Lionnais, was fascinated by all this. In 1983, he wrote a paper in which he called 2, 3, 5, 11, 17 and 41 “lucky numbers of Euler". Le Lionnais is better known, though, for co-founding the Oulipo literary movement, whose adherents write using constraints, like not using the letter “e" (George Perec’s novel A Void is a famous example); and he also co-founded Unesco’s Kalinga prize for the popularization of science, set up with a donation from Biju Patnaik.

But, and speaking of “e", there’s more; especially since we’re in April. Heegner numbers got a measure of fame in that month in 1975, courtesy the irrepressible Martin Gardner. In his Mathematical Games column in Scientific American, Gardner claimed that if you raise e to the power of the square root of the largest Heegner number multiplied by p (pi)—that is, calculate epÖ163 —you will get an integer.

Gardner wrote that Srinivasa Ramanujan had, in a 1914 paper, predicted this result.

What’s the big deal here, you ask? Anyone who knows anything about e, p and the square root of 163 would have been astounded by this claim. All three are irrational numbers, and mathematicians know that such manipulations of irrationals cannot produce an integer. Had Gardner, or Ramanujan before him, stumbled on some incredible mathematical result, just as incredible as a formula for primes?

Sadly, no! In fact, as Gardner admitted in July that year, this was an April Fool’s joke he pulled on his readers. Ramanujan never mentioned this number. More important, it isn’t an integer anyway; yet the truly astonishing thing is how agonizingly close it comes to being an integer. For if you calculate epÖ163, you’ll get 262,537,412,640,768,743.9999999999992511…, which is essentially indistinguishable from 262,537,412,640,768,744.

Another French mathematician, Charles Hermite, had first noticed this peculiar property of the Heegner number 163 — in 1859, long before either Heegner or Ramanujan was even born. Yet, Gardner’s April Fool hoax had its effect. Canadian mathematician Simon Plouffe, who broke a world record in 1975 by memorizing and reciting 4,096 digits of π, labelled epÖ163 “Ramanujan’s constant". That name has stuck.

And as you probably have guessed, the Heegners 19, 43 and 67 give us similar results. Manipulate them using e and π in the same way, and you’ll get numbers that are nearly as agonizingly close to an integer as Hermite’s number — Ramanujan’s constant — above. But also, the integer in each case is the sum of 744 and the cube of a particular number.

That is:

• epÖ19 is 885,479.777… which is nearly 744 + 963, or 885,480.

• epÖ43 is nearly 744 + 9603.

• epÖ67 is nearly 744 + 52803.

• epÖ163 is nearly 744 + 640,3203.

(Note: the Heegner numbers 7 and 11 generate numbers that are not as visibly close to an integer as the others do, but the same fiddle with 744 and a cube applies).

Also, while it’s true Ramanujan did not find the number Plouffe named after him, it’s possible he might eventually have done so. Because he was playing around with e and p and square roots too, and had discovered that epÖ58 is also agonizingly close to an integer. In fact, to 744 + 29083.

In fact too, 22, 37, 88, 148 and 232, manipulated with e and p like this, lead to similar numeric coincidences involving 744, even though none of these is a Heegner number.

Perhaps these numbers and manipulations make your head spin as they do mine. What I wondered, though, was why there are all these close shaves with integers. There is an explanation, but it involves such mathematical concepts as modular forms and the Monster simple group, and I remember a mention of this particular Monster I once ran across: it is “best understood in terms of the symmetries of a 196884-dimensional algebraic object".

Which is to say, the close-shave explanation is a little beyond this column’s mathematical ambitions. So I’ll leave it there.

But there’s still more here. With p being an intimate part of these goings-on, and π being the irresistible number it is to many mathematicians, it’s no surprise that they have found formulae to generate p using some of all this number juggling. In 1988, New York University’s Chudnovsky brothers, David and Gregory, announced formulae for p that involved the largest Heegner number, 163. They and others have used these to set world records for calculating the digits of p. That includes the most recent record, announced on p-day (14 March) this year: Emma Haruka Iwao, a Google employee in Japan, calculated p to 31,415,926,535,897 (that’s more than 31 trillion) digits.

Maybe you notice how that number itself is related to p.

Now to my knowledge, there’s no real-world application in which you will ever need to know, let alone rattle off, those 31 trillion digits. Calculating them is really just a task to test powerful computers with—Iwao’s machines, for example, took 121 days to run the computation. But even so, it is a reminder of the deep and subtle connections in mathematics: from prime numbers via 744 to the endless digits in p, who would have thought?

Not only that: who would have thought this exploration of the charms of Heegner numbers would take in names from Gauss to Gardner, Chudnovsky to Ramanujan, Euler to Patnaik? Not forgetting e, both number and letter shunned by Oulipo adherents?

And by now, I hope you’re wondering, as any mathematician would, “why 744?" Why does it appear in the equations above? Of course there is an explanation, and I found it succinctly stated in the same place I found the Monster mention above: it is the “coefficient of the linear term in the q-expansion of the j-invariant".

Reading which brought to mind the final name in this story: Amitabh Bachchan. If you remember the 1970s classic Amar Akbar Anthony, you will remember Bachchan reciting an immortal line that starts with these words: “You see the coefficient of the linear…"

Hmm. Does Bachchan have a subtle link to 744?

Well, his immortal line was this: “You see the coefficient of the linear is juxtaposition by the haemoglobin of the atmospheric pressure in the country!"

And the atmospheric pressure you’re feeling right now, expressed in millimeters, is about 760.

Close enough, I’d say, to 744.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun