I’m the urban spaceman, baby; I’ve got speed.
I’ve got everything I need.
− Neil Innes, “I’m the Urban Spaceman” (Bonzo Dog Doo-Dah Band)
There’s an episode1 of a science-fiction television series in which space travelers land on a planet peopled by their own descendants. The descendants explain that the travelers will try to leave the planet and fail, accidentally stranding themselves several centuries in the past. Armed with this knowledge, the travelers can try to thwart their destiny; but are they willing to do so, if their successful escape would doom their descendants, leaving the travelers with the memory of descendants who, thanks to their escape, never were?
This is science fiction, but it’s also math. More specifically, it’s proof by contradiction. As Ben Blum-Smith recently wrote on Twitter: “Sufficiently long contradiction proofs *make me sad*! When you stick with the mathematical characters long enough, you start to get attached, and then they disappear, never to have existed in the first place.”
This will be an essay about things that seem as if they exist but which, when you study them deeply enough, turn out not to exist after all.
Odd perfect numbers are a good example, unless they aren’t. A perfect number is a number that is the sum of its own divisors (not including the number itself); so for instance 6 is perfect because 6 = 1+2+3 and 28 is perfect because 28 = 1+2+4+7+14. Are there any odd perfect numbers? Nobody’s found one yet, but we’ve learned a lot about odd perfect numbers over the past five centuries; someday we may know so much about them that we’ll be able to conclude that there aren’t any. Or maybe our dossier of the properties of odd perfect numbers will eventually guide us to one.
This uncertainty is endemic to math research; often we don’t know whether our efforts will result in a proof or a counterexample, and it’s often a back-and-forth process. Douglas Adams, in Life, the Universe, and Everything, wrote that the way to fly is to throw yourself at the ground and miss. In an almost, but not quite, entirely similar fashion, the way to find a proof is to throw yourself into constructing a counterexample and fail, and the way to find a counterexample is to throw yourself into constructing a proof and fail.
There are horror stories in math about people who didn’t throw themselves hard enough and got thrown by others, typically during a thesis defense. The apocryphal example I heard when I was in graduate school was about some hapless Ph.D. candidate who wrote a dissertation about Hölder-continuous functions with exponent α > 1. He proved all sorts of regularity properties satisfied by these functions (“They’re infinitely differentiable!” “Their power series have infinite radius of convergence!”), only to find out during his thesis defense that all such functions are, in fact, constant. That’s not quite the same as finding out that the mathematical objects you’ve been studying don’t exist, but it’s almost as bad.
Ben Blum-Smith’s tweet was prompted by his reading of I. Martin Isaacs’ book Finite Group Theory. Ben sent me an annotated summary of Isaacs’ 5-page proof of Burnside’s pq-theorem, and I won’t go into technical details because most of you don’t want me to, but the important thing is that Ben’s account included his own emotional arc alongside the logic of the proof; I got a vicarious sense of his elation at the midpoint of the proof and his all-the-greater sense of deflation at the end. Metaphorically speaking, in the middle of the proof the clouds part and sunlight illuminates a hitherto hidden landscape with marvelous structure, and then the light gets even brighter, but at the end the light gets really bright and the seas boil away and the sun explodes and the whole planet is obliterated, The End. And the landscape you saw wasn’t even really there; it was a smudge on your glasses, a flaw in your vision, a consequence of your human inability to perceive the whole truth at once.
My favorite example of something that seemed like it might possibly exist, until someone proved that it didn’t, is the tenth Heegner number, named after Kurt Heegner (“HAY-g’nur”), the person who proved that it didn’t exist. The proof is far too technical for me to include here, so let me start with a simpler example of something that doesn’t exist; historically speaking, it may be the first number that ever never was.
THE LAST PRIME
It’s often said that Euclid proved that there are infinitely many primes, but that’s an anachronistic paraphrase, since the Greeks didn’t have the modern notion of infinite collections. It’d be more in keeping with the Greek mode of thought to say that Euclid proved that there is no last prime.2
The famous argument goes like this: Assume there’s a last prime. Multiply together all the primes from the first to the last, obtaining some big number N, and consider the number N+1. N+1 can’t be divisible by any of the primes, because N is divisible by all of the primes, and two consecutive integers can never have a prime factor in common.3 So either N+1 is a prime number that you missed or it’s a product of smaller primes that you missed, but either way, your list of primes is incomplete.4
See what happened there? We assumed that there is a last prime and then showed that this assumption contains the seeds of its own destruction; the assumption devours itself, ouroboros-like, until nothing is left but our memory of its self-undoing.
I avoided giving the last prime a name so you wouldn’t get too attached to it, but even if I’d named it, you probably wouldn’t have started to develop feelings about it; the proof of its nonexistence was too short. But when a proof gets long enough, and you spend enough time wading through a logical swamp in search of the logical contradiction that dissolves the swamp, you can start, on emotional level, to buy into the very fiction you’re trying to subvert. And this can be even more true on longer time-scales, where the suspense lasts not hours but decades or centuries.
Can we find a formula for primes? It depends on what kind we want. If we want a simple, practical formula that, when you plug in n, gives a formula for the nth prime, then it’s been known for a long time that no such formula exists.
However, if you scale back your ambition and search for simple functions that generate lots of primes, there’s a great example devised by the 18th century mathematician Leonhard Euler: the polynomial n2 − n + 41. It takes on prime values for all n between 1 and 40. A nice way to visualize this phenomenon is with an “Ulam spiral” (credited to the Polish mathematician Stanislaw Ulam):
We put the number 41 into a box in an infinite square grid and travel outward in a square spiral, labelling the boxes with successive integers as we go. Then there’s a long diagonal stretch of boxes containing nothing but primes (shown in blue). This blue streak of primes is so long that the picture isn’t big enough to contain it!
As Euler may have noticed, there are other numbers k that are “41-ish”, in the sense that, for all n between 1 and k−1, the values taken on by the polynomial n2 − n + k are all prime. Specifically, the values 1, 2, 3, 5, 11, 17, and 41 all have this property. And Euler may have wondered whether there were other, larger values of k beyond 41 that had this property.
Euler’s observation took on a new significance in the 19th century, when Carl Friedrich Gauss initiated the study of number systems obtained from the rational numbers by throwing in certain irrational numbers. These are called algebraic number fields, and we often represent a typical algebraic number field by the letter K.5 The simplest algebraic number fields are the ones you get by throwing in the square root of d, where d is some rational number that, like −1 or 2, doesn’t have a rational square root. This is written as K = ℚ(sqrt(d)) (as opposed to plain old K = ℚ, the field of rational numbers); it consists of all numbers of the form a + b sqrt(d) where a and b are ordinary rational numbers. Such a K is called a quadratic number field. We may assume without loss of generality6 that d is a “squarefree” integer, that is, an integer whose prime factorization contains no repeated prime factors (unlike, say, 4 or 8; ℚ(sqrt(4)) is just ℚ, while ℚ(sqrt(8)) is the same field as ℚ(sqrt(2)), since sqrt(8) equals 2 sqrt(2)).
One important quantity associated with a number field K is a positive integer h called the class number of K. When h=1, the Fundamental Theorem of Arithmetic (asserting uniqueness of factorization into primes) holds in K just as it does in ℚ (even though the primes in K are different from the primes in ℚ). When h>1, the Fundamental Theorem of Arithmetic fails, because some of the numbers in K “want” to be factored into primes belonging to a bigger field L, called the (Hilbert) class field of K that in a certain sense is h times bigger than K.7 (Gauss didn’t know about class fields, and his understanding of class numbers was more based on quadratic forms than on quadratic extensions of ℚ, but the key insight was his.)
In 1801, Gauss observed a curious dichotomy. There are lots8 of positive squarefree values of d for which the (real) quadratic field ℚ(sqrt(d)) has class number equal to 1 and hence has unique factorization, but he found only a few negative squarefree values of d for which the (imaginary) quadratic field ℚ(sqrt(d)) has class number equal to 1, specifically, −1, −2, −3, −7, −11, −19, −43, −67, and −163.
The nine numbers 1, 2, 3, 7, 11, 19, 43, 67, 163 are often called Heegner numbers nowadays, although the name is a little misleading; naming this set of numbers after Heegner is like naming a species not after the naturalist who discovered them but after the naturalist who declared the species extinct!
There’s a hidden connection between the Heegner numbers and the prime-generating polynomials Euler studied. To see the connection numerically, throw out the first two numbers in the list, and just keep the rest:
3, 7, 11, 19, 43, 67, 163
Now add 1 to each of them:
4, 8, 12, 20, 44, 68, 164
And now divide each number by 4:
1, 2, 3, 5, 11, 17, 41
These are precisely the values k for which n2 − n + k takes on only prime values for n = 1, 2, …, k−1.
This is not an accident. Georg Rabinovitch showed in 1913 that for d = 4k−1, the field ℚ(sqrt(−d)) has class number 1 (or, equivalently, the unique factorization property) if and only if the polynomial n2 − n + k gives prime values for n = 1, 2, …, k−1.
Gauss conjectured that ℚ(sqrt(−163)) was the last of the imaginary quadratic fields with class number 1 — that is, as we would put it nowadays, that 163 is the last Heegner number — but he wasn’t able to prove it. This was Gauss’ “class number 1 problem”, and it stayed unsolved for more than a century and a half. It’s equivalent to the conjecture that Euler’s prime-generating polynomial n2 − n + 41 is the last polynomial of its kind.
Gauss also conjectured that for each positive integer h, there are only finitely many imaginary quadratic fields with class number h. This was Gauss’ “class number problem” (not to be confused with his class number 1 problem).
Half a century before Rabinovitch, in 1859, the French mathematician Charles Hermite discovered an amazing formula involving Heegner numbers, the constant e, the constant π, and a mathematical function called the j-function. A consequence of Hermite’s formula was that when k is 43, 67 or 163, e raised to the power of π sqrt(k) is weirdly close to an integer.9 Indeed, e to the power of π sqrt(163) equals
262 537 412 640 768 743 . 999 999 999 999 25 …
It’s worth stressing how striking this near-miss is; it’s rare for irrational numbers to come so close to being whole numbers (aside from numbers like sqrt(999999), for which the reason for the small difference is obvious).
I wonder if Hermite was as impressed by his formula as I am. I like to imagine him geeking out at some École Polytechnique social event, telling everyone about his wonderful new result; and then I imagine his engineer colleagues saying “Well, if the number is so big, and the discrepancy is so small, then for all practical purposes it might as well be a whole number, so what’s the big deal?”10
Martin Gardner exploited the near-miss in his April Fool’s Day column of 1975, in which he claimed that e to the power of π sqrt(163) was exactly equal to 262 537 412 640 768 744 (see the References). Unfortunately this hoax had the lasting effect of introducing the memorable caconym “Ramanujan’s constant”, which has stuck, making many people mistakenly think that this result was due to Ramanujan.
If there were a tenth Heegner number, like 163 only bigger, the work of Hermite shows that there’d be another numerical coincidence, like the one involving e to the power of π sqrt(163) but even closer. And if there were an eleventh Heegner number, there’d be an even more stupefyingly close coincidence. Hermite (as far as I’m aware) didn’t speculate about whether there were more numbers of this kind, but I’m pretty sure that if he gave the matter any thought, he would’ve agreed with Gauss that there probably aren’t any more.
Yet, for all Hermite and his contemporaries could prove, there might be infinitely many more numbers, each giving a more amazing near-miss than the one before.
HEILBRONN AND LINFOOT
In 1934, Hans Heilbronn solved Gauss’ class number problem when he proved that for each positive integer h, there are only finitely many imaginary quadratic fields K with class number h.
This had implications for Gauss’ class number 1 problem; it showed that there are only finitely many imaginary quadratic fields with class number 1. That is, Heilbronn’s work showed that even if 163 wasn’t (as Gauss had conjectured) the last Heegner number, there were at most finitely many more that Gauss hadn’t known about. Maybe none, maybe one, or maybe two or more?
That ambiguity didn’t last long. Working with Edward Linfoot in that same year, Heilbronn proved that there is at most one more Heegner number beyond the nine that Gauss and Hermite knew about.11 The proviso “at most one” invites us to reify the tenth Heegner number in a way that Heilbronn’s earlier result doesn’t. This says more about the human mind than it does about mathematics. We crave concreteness, and “at most one” offers the prospect of specificity, though with no promises. Such a theorem might have tempted some people to step out further and further, chiliad by chiliad12, in pursuit of an elusive tenth Heegner number that never came into view but could, for all one knew, be around the next corner.
I’m not saying anyone looked for a tenth Heegner number; there was no reason to doubt Gauss’ opinion. But the long odds against there being a tenth Heegner number might have tempted some number theorists with a contrarian streak to check a few far-out chiliads, just in case.
Although he’s largely unknown to nonmathematicians (and to most mathematicians outside of the field of number theory), Kurt Heegner was one of the greatest amateur mathematicians of the 20th century. He was a radio engineer who pursued number theory as a hobby, and his great achievement was his solution, in 1952, of Gauss’ class number one problem.
Unfortunately, Heegner’s proof was regarded as incomplete when he published it and it was not studied by many number theorists during his lifetime. In the 1960s, mathematicians Alan Baker and Harold Stark found proofs of their own. When Stark took a look at Heegner’s proof, he saw that Heegner’s proof was essentially valid, and that the gaps in the proof were easily filled; Heegner most likely could have supplied those missing steps had anyone shown any interest in his approach and asked him to clarify those steps.
In any case, for hundreds of years the question of whether there was a tenth Heegner number was an open question, and for 20 or 30 years, the question was lent extra piquancy by the knowledge that there was no eleventh. The theorem of Baker, Heegner, and Stark settled the matter. The ninth Heegner number is the last of its kind; there is no tenth.
In some ways, the fact that the phrase “the tenth Heegner number” ever cropped up to begin with was the result of historical happenstance. As Stark pointed out in his 1969 paper, if the nineteenth century mathematician Heinrich Martin Weber had taken a closer look at results in his own 1895 treatise Lehrbuch der Algebra, he might have solved the class-number one problem at the end of the nineteenth century. And if Weber had found a complete solution of the class number one problem before Heilbronn and Linfoot could find a partial one, then the prospect of a tenth-and-final Heegner number would never have achieved the level of intrigue that it possessed for two or three decades.
If other intelligent minds exist in the universe, and they do math that’s intelligible to us, there are likely to be both satisfying similarities and striking differences. If aliens care about primes, they’re bound to discover that there are infinitely many primes. And if they care about number fields and class numbers, they’ll care about imaginary quadratic fields with class number 1, and they’ll discover Heegner numbers, and they’re bound to discover that there are only nine of them. But will they ever be at a stage of knowing that there are at least nine and at most ten? It seems to me fairly unlikely that the twists and turns of other planets’ mathematical histories will reproduce this particular circumstance of ours. So if anyone’s going to mourn the tenth Heegner number, it’ll have to be us.
I’m the urban spaceman, baby; here comes the twist:
I don’t exist.
Thanks to John Baez, Ben Blum-Smith, Veit Elser, Sandi Gubin, Keith Lynch, Kerry Mitchell, Ken Ono, Ben Orlin, and Evan Romer.
Next time: Flip your students, flip yourself.
#1. “Children of Time”, Star Trek: Deep Space Nine.
#2. It’d be even more in keeping with the original to phrase the result positively: “Given any finite list of primes, there’s a prime not on the list”. For a faithful rendering of Euclid’s proof into English, see David Joyce’s translation.
#3. Just last week my twelve-year-old warmed my heart by asking “When an odd and an even number are next to each other, can they ever have a common factor?” The answer is no, for a simple reason: Whenever d is a common factor of two numbers, it’s also a factor of their difference, so any prime that divided both N and N+1 would have to divide (N+1) − 1 = 1, which is impossible.
#4. Say for instance that your list consists of the primes 2, 3, 5, and 7. Then N is 2×3×5×7=210 and N+1 is 211, a prime you missed. Or, say your list consists of the primes 2, 11, and 17. Then N is 2×11×17=374 and N+1 is 375, which is a product of 3’s and 5’s, both primes missing from your list.
#5. What English-speaking mathematicians call a “field”, German-speaking mathematicians call a “Körper”, or body; hence the K. I’ve sometimes wondered whether the difference in metaphor reflects, or perhaps imposes, subtly different ways of thinking about number systems.
#6. If you ask “Why assume d is squarefree? Why are we excluding ℚ(sqrt(75)), for instance, just because 75 has a repeated factor of 5?”, then the answer is “We’re not excluding that field; we’re just including it under a different name, namely ℚ(sqrt(3)).”
#7. For instance, K = ℚ(sqrt(−15)) has class number h(K) = 2, and the right field for factoring general elements of K is the larger field L = ℚ(sqrt(−3),sqrt(5)). L is “twice as big” as K in the same sense that field of complex numbers is “twice as big” as the field of real numbers.
#8. Gauss conjectured that there are infinitely many squarefree positive integers d for which ℚ(sqrt(d)) has class number equal to 1, but this famous problem is still unsolved.
#9. You may be wondering why there are near-misses for the larger Heegner numbers but not the smaller ones. When k is one of the smaller Heegner numbers, e raised to the power of π sqrt(k) still has a series expansion as an integer plus smaller correction terms, but the first correction term is about .22 for k = 19 and is even bigger for k < 19, so we don’t get a near-integer.
#10. If any of you know more about how Hermite felt about this result, please let me know in the Comments!
#11. The article by Ayoub listed in the references gives a clue about how one might do this. The proof uses L-functions, which belong to the same province of math as the Riemann zeta function mentioned earlier, namely analytic number theory. Ayoub shows, for a certain constant C, that if d and d‘ are both bigger than C and have the property that the fields ℚ(sqrt(−d)) and ℚ(sqrt(−d‘)) both have class number 1, then d=d‘. Then one can conclude that at most one Heegner number exceeds C.
#12. A chiliad is a range of a thousand consecutive integers. It’s a natural chunk of Numberland to survey in search of numerical quarry. It’s said that when Gauss had some time to kill, he’d move on to the next chiliad and find the primes there.
Martin Gardner, “Six Sensational Discoveries”. Chapter 10 of “Time Travel and Other Mathematical Bewilderments”.
Aled Walker, “Heegner’s Solution to the Class Number 1 Problem”.
Wikipedia, “Heegner number”.
Wikipedia, “Class number problem”.
Wikipedia, “Stark-Heegner Theorem”.