The recent proof of the weak Goldbach Conjecture calls for revisiting this gem, originally published in the 1970s in Manifold and later collected in Seven Years of Manifold: 1968-1980.
An odd evening
by Ian Stewart
As dusk settles gently over the undulating English countryside we find our hero Rosen Crantz, research student, discussing his latest ideas with his supervisor, Prof Guilden Stern, a none-too-successful number-theorist.
Crantz: Guilden, I’m stuck on my research problem.
Stern: What, the one about prime numbers?
Crantz: Yes. I was going to prove it for each prime number in turn, using that paper of Randy and Hartlisnujam…
Stern: You mean A Complete List of all Prime Numbers, Journal of Infinity, volumes 173 onwards?
Crantz: Yes, but they’ve only published the even primes so far – I think they got stuck somewhere.
Stern: I had a letter form Hartlisnujam a few weeks ago. He said they’d started off well with 2 – that’s prime of course – and they decided to run through all the even numbers first in the hope of finding some more. He said they’d got up to about 8346318892823676 so far but hadn’t found any.
Crantz: Perhaps there aren’t any other even primes?
Stern: But what about that theorem of Dirichlet’s you know, the one that says that there are an infinite number of primes in any arithmetic progression? The even numbers form an arithmetic progression, don’t they?
Crantz:I guess so. I’ve forgotten most of what I did at school. It’s very puzzling.
Stern: Perhaps Dirichlet made a mistake? He did with his principle, you know.
Crantz: Wasn’t that Riemann? Anyhow, it seems unlikely. Maybe we could prove there exist infinitely many even primes?
Stern: By modifying Euclid’s proof for arbitrary primes, you mean?
Crantz: Exactly. We’ll work with just even primes and see what happens. Suppose there’s only a finite number…
Stern: We can miss out 2, we know about that…
Crantz: So let’s suppose that there are only finitely many even primes greater than 2, say p1, …,pn. Now what? Euclid forms p=p1…pn+1 and…
Stern: That won’t work. It’s odd.
Crantz: Very odd.
Stern: Ha. So why not define p=p1…pn+2?
Crantz: OK. Then p is even so it must be divisible by some even prime, say q. And q can’t be any of the p’s since they leave a remainder 2 when you divide p by them…
Stern: …and it can’t be two, since if 2 divides p then it divides p1…pn, so it divides one of the p’s… but that p is prime and greater than two so it can’t be divisible by two.
Crantz: So q is an even prime not equal to 2,p1,…,pn,…
Stern: Contrary to our assumption. So there must be an infinite number of even primes altogether.
Crantz: I guess that does it. Dirichlet was right after all.
Stern: I’ll write to Hartlisnujam about it.
Crantz: I wonder if it’ll help my problem?
Stern: What is your problem?
Crantz: Uh… well… I think my girlfriend is…
Stern: Your research problem.
Crantz: Oh, yeah. It’s a sort of converse to Goldbach’s Conjecture.
Stern: You mean “every even number is the sum of two primes?”
Crantz: Yes. I want to prove that every prime is the sum of two even numbers. You see, if I could prove that, then…
Stern: But it’s false, surely? What about 3? If 3 is the sum of two even numbers, then one of them is 2… so the other is 1. And that’s odd.
Crantz: Very odd.
Stern: Ha. You need extra hypotheses. Why not assume your prime is even?
Crantz: I thought of that. But suppose we take an even prime q and assume that q=x+y where x and y are even – say x=2u and y=2v. Then q=2(u+v) so 2 divides q. But q is prime – contradiction.
Stern: So that disproves it for even primes.
Crantz: Does it? I never realised…
Stern: Which means you only need look at odd primes.
Crantz: But I can’t wait for Randy and Hartlisnujam to get to them…
Stern: Well, anyway you’ve disposed of half the possible cases.
Crantz: Plus 3, which you did.
Stern: Then write it up and publish it. That way, if you do work out the odd ones, you get two papers out of it.
Crantz: I thought they weighed publications, rather than counting them?
Stern: No, that was before they started printing Mosaic on stone tablets. No; five papers and you’re a lecturer, fifteen a senior lec-
Crantz: Wait! Wait! Where in the proof have we assumed that q is even?
Stern: Oh, where we – no. We didn’t. We haven’t! The same proof works for odd primes too!
Crantz: I can see it now! Falsity of the Converse Goldbach Conjecture by R.Crantz –
Stern: And G. Stern…
Crantz: Yes. We could publish it in the Notices…
Stern: The Journal…
Crantz: The Bulletin…
Stern: The Proceedings…
Crantz: The Transactions…
Stern: The Annals…
Crantz: … The Ivanov Gos. Ped. Inst. Uc. Zap. Fiz. -Mat. Nauki –
Stern: (thumping him on the back) Nasty cough you’ve got there.
Crantz: What a reference!
Stern: Fame! Fame! Oh, wait till I see Stevie Smale…
Crantz: We can present it at the International Congress of Mathematicians. We might even get a Fields Medal.
Stern: Two Fields Medals.
Crantz: I’ll be a professor in no time. They make thousands, you know. Absolutely rolling in it. I hear one of them recently sold his thirteenth century cellar…
Stern: No! Really?
Crantz: And I won’t even have to write thirty-one papers and two –
Stern: I could do a lecture tour in the USA!
Crantz: A sort of Malcolm Muggeridge?
Stern: Not exactly; more a Charles Dickens or a – what was that American chap’s name?
Stern: No, I’ll hire a car.
Crantz: And I could do a tour of Paris – lunch at the Sorbonne, dinner at the Institut – I might even get to meet Bourbaki! Yes! Yes! (He pauses, suddenly puzzled.) Wait a minute. What about 2?
Stern: What of it? Go on, go on!
Crantz: 2 is prime. 0 and 2 are even.
Stern: Oh, BOTHER!
Crantz: Maybe we could patch it up…
Stern: But where have we assumed things are non-zero? I don’t see it. It’s odd.
Crantz: Very odd.