This is a followup to my last post. I promised some more problems in which there is an initial pattern that’s wrong. Here is one more. It is not nearly as good as the points-on-a-circle problem I discussed before, for reasons I’ll say below. But I’m brainstorming here, and hope you’ll join me, so anything’s better than nothing. (And actually I think it’s a cool problem in its own way.) Thanks Kate, jd2718, and Gilbert for contributing ideas so far.
The problem involves seeking a simple formula that produces prime numbers only. As you probably know, in spite of centuries of research no such formula is known to this day. There is some fun history around this. For example, Fermat believed that 2^k + 1 was prime whenever k was a power of 2. It is prime for k=1, 2, 4, 8 and 16. However for k=32, the number is 4,294,967,297 which was found by Euler to be equal to 641 * 6,700,417. Now, even in the age of computers, no other prime of the form 2^k + 1 has yet been found. Of course, I’d avoid putting kids in the position of having to calculate 4,294,967,297 or to show that it’s not prime.
Anyway, the idea is to get a class engaged in a search for such a formula. My idea for how the lesson goes would be to try out some examples with them to show them what is being sought. Like, maybe 4n + 3 which equals 3, 7, 11 for n = 0, 1, 2 but then fails for n = 3, or, starting with p_1 = 2 and then recursively doubling and adding 1 which gives you 2, 5, 11, 23, 47 and then the next one fails. (A closed form for this last one would be
3*2^n – 1 for n = 0, 1, 2, 3, 4.) This second formula is a good replacement for Fermat’s conjecture, because it gives you 5 primes before it fails, just like Fermat’s conjecture, but the primes are a reasonable size and the one that fails (95) is obviously not prime. Anyway, once they understand what’s being sought, the problem is to find such a formula. They will totally fail and they have no tools that will help them, so don’t let them stew too long. Then, show them a very pretty creation of Euler’s:
n^2 – n + 41. This quadratic polynomial is (amazingly) prime for n=0, 1, 2, …, 40. So there’ll be some initial excitement as this one seems to answer the question. But actually, it can’t possibly answer it. And the class may be able to see that the n=41 case will fail without actually doing the calculation. Even if the calculation is needed, it can still lead to a cool conversation.
Now this isn’t as rich as the points-on-a-circle problem because the inordinate primality of n^2-n+41 is sort of a mathematical accident; there isn’t a rich story behind it (at least not one I’ve ever heard), so once the pattern is noticed and then broken there’s nowhere to go. But it does at least give students some experience of the fact that if a rule holds for small cases, it doesn’t mean it always holds. And the breakdown at n=41 is accessible to reasoning alone, without calculating. So it’s a win for the power of mathematical reasoning over raw pattern-noticing.
Other ideas in this vein? (Problems where there is a “obvious” or “apparent” pattern or conclusion that is actually wrong?)