Misremembering Goldbach’s Conjecture

It's the holiday, so a shorter, lighter blogpost today, and only one reflection question. I hope you are having a good break!

When I went on Craig Barton’s podcast for the first time (Note 1), he asked me (as he asks all his guests) to recount a ‘favourite failure’ - a situation where things didn't go to plan. I had so many to choose from that I had plenty of ideas leftover after the episode, so I thought I’d relate another one here… 

This is about the time when I misremembered Goldbach’s Conjecture, which states:

Every even integer greater than 2 is the sum of two primes.

Unfortunately, for some reason, I misremembered it as:

Every integer greater than 2 is the sum of two primes.

If I had taken a moment to reflect on this, I would have realised that this obviously couldn’t be right, but it was one of those situations where I was distracted or ill or something (I can’t remember the specifics of my excuses!). And so I noticed nothing and carried on...

I wanted my Year 8 class (age 12-13) to work on something a bit exploratory and to understand the notion of a ‘counterexample’ – and also get in a bit of incidental practice on recognising prime numbers, which we had just been working on. So, this seemed like a good way to address all of that.

So, I told the class that Goldbach’s Conjecture was one of the best-known unsolved problems in all of mathematics, and I explained what a counterexample was. No one knows how to prove that Goldbach’s Conjecture is true, but, if it is false, all it needs is one counterexample to demonstrate that. A single counterexample can do a great deal of work!

The students seemed interested in this:

“Would we be famous if we found a counterexample?”
“Sure!”

There was immediately a bit of confusion about the number 3, which should have alerted me to the fact that something was wrong. Some pupils had written $3 = 1 + 2$, but others were – correctly – saying that 1 is not a prime number, in which case 3 would be a counterexample. I knew that 1 did used to be considered as a prime number (see Foster, 2016), so I thought perhaps this was just a historical glitch, so I decided that we would say “every integer greater than 3”, rather than 2, to avoid that problem.

And so the students began work. Of course, I knew very well that they would not find a counterexample, since all numbers at least as far as $4 \times 10^{18}$ have been checked (Note 1). If ever a teacher knew ‘the right answer’, I knew that the right answer here was that there would be no counterexample today!

The students began work on their own or in pairs, writing (at least, those working more systematically!) things like this:

$4 = 2 + 2$
$5 = 2 + 3$
$6 = 3 + 3$
$7 = 2 + 5$
$8 = 3 + 5$
$9 = 2 + 7$
$10 = 5 + 5$
$11 = …$

I walked around casually observing what was going on, my mind drifting a little, perhaps. I engaged in some discussion with students about who Goldbach was, why prime numbers matter, and so on, in quite a relaxed way. This was basically routine practice with primes in a more interesting context (a kind of mathematical etude, see Foster, 2018).

I gradually became aware that I could hear the number 11 being muttered quite a bit.

Then a few people started to say that they had found a counterexample, and it was 11. I decided that this would be a good opportunity to stop everyone and highlight the importance of ‘being systematic’. There's 'being systematic' in the sense of choosing your numbers according to some pattern, rather than haphazardly, but there's also 'being systematic' when you check each number. It’s easy to think you have found a number which can’t be made by summing two primes, and it may just be because you haven’t thoroughly checked all the possibilities. You haven’t managed to find a pair of primes that sum to 11, but that doesn’t mean that there isn’t one. The only way to be sure is to be systematic and check all the possibilities in such a way that you can be sure that you haven’t missed any. “Go back and check – be systematic – make sure you haven’t missed a possibility!” All good advice, to be sure.

I vividly remember the moment that one student came to the board and said, essentially:

Look, the only possibilities for 11 are:

1 and 10, but neither is prime 
2 and 9, but 9 isn’t prime 
3 and 8, but 8 isn’t prime 
4 and 7, but 4 isn’t prime 
5 and 6, but 6 isn’t prime 
So, 11 is a counterexample. 

Ordinarily, I would have been very pleased with such a proof by exhaustion. But, I remember staring at the board thinking, “What am I missing?” Even if we included 1 as prime, it would have to go with 10, which had certainly never been prime in anyone’s book!

As I tried to figure out what was going on, the class became more excited at my puzzlement:
“We’ve done it! We’ve solved this big maths problem – and it wasn’t even that hard!”, “Are we going to be on the news, sir?”, “Maybe no one ever bothered to check 11 because they assumed someone else had already checked it? Sometimes it’s the easy things that get missed!”, etc. 

Obviously, if there were a counterexample, it was going to be considerably higher than 11. So, feeling desperate, I now Googled “Goldbach’s Conjecture”: “Every even integer greater than 2 is the sum of two primes.” ‘Even, even, even!’ (Having computers connected to the internet in every classroom has to be one of the great pedagogical advances of recent decades.)

Of course, with hindsight it is very clear that the only way to make an odd number by summing two integers is if one of the integers is odd and the other one is even. And the only even prime is 2. So, the only way my version of Goldbach’s Conjecture could be true is if every odd number were 2 greater than a prime. This is equivalent to saying that every odd number is prime, and although it is true that (almost, with the exception of 2), every prime number is odd, the reverse is not the case. This is why we had the problem with 3, because 1, which is $3-2$, is not prime. But then 5, 7 and 9 all have primes 2 less than them, so everything seems fine for a while, but then 11 doesn’t, because 9 is not prime, and that’s why it had appeared to be a counterexample. So, at least there was something mildly interesting to understand in relation to my mistake. Obviously, a counterexample to one conjecture is not necessarily a counterexample to a different conjecture.

There was understandably limited enthusiasm now for going back and checking for counterexamples to the real Goldbach Conjecture. It felt like the moment had passed, and perhaps the objectives of understanding what a counterexample is and gaining facility with primes had been accomplished more or less anyway.

I reflected afterwards on the strange feeling of seeing the student's apparently flawless proof and yet not believing it – the feeling that ‘there must be something wrong even though I can’t see what it is’. However rational we might aspire to be about mathematics, we are influenced by more than merely logical arguments. I was quite sure that the students must have made a mistake and omitted a possibility, and I was reluctant to believe even the very simple mathematics of their proof until I had appreciated the wrong assumption that I had begun the whole lesson with.

Question to reflect on

1. Do you have any 'armchair responses' (AssocTeachersMaths, 2020; Foster, 2019) to my ‘favourite failure’ or to any of your own?

Notes

1. You can listen to the episode here: http://www.mrbartonmaths.com/blog/colin-foster-mathematical-etudes-confidence-and-questioning/ 

2. See http://sweet.ua.pt/tos/goldbach.html

References

AssocTeachersMaths. (2020, July 13). Armchair Responses to Classroom Events - with Colin Foster [Video]. YouTube. https://youtu.be/L0ovhillL0c

Foster, C. (2016). Questions pupils ask: Why isn’t 1 a prime number? Mathematics in School, 45(3), 12–13. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Why%20isn't%201%20a%20prime%20number.pdf

Foster, C. (2018). Developing mathematical fluency: Comparing exercises and rich tasks. Educational Studies in Mathematics, 97(2), 121–141. https://doi.org/10.1007/s10649-017-9788-x

Foster, C. (2019). Armchair responses. Mathematics in School, 48(3), 26–27. https://www.foster77.co.uk/Foster,%20Mathematics%20in%20School,%20Armchair%20responses.pdf