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So I'm doing a mathematics education extension for my current undergraduate maths course, and for one bit of the final assessment we're asked to create a detailed lesson plan on the (strong) Goldbach Conjecture - that any even number is the sum of two primes.

Part of the task is to make it a flexible enough plan that it's fairly independent of age and time allotted. We're also assuming the students know at least what primes are.

I've got a few ideas (listed below), but I was just wondering if there's any interesting/novel ways of explaining or exploring the Goldbach Conjecture at primary or secondary student level? There seems to be some interesting symmetry/geometric number line intuition one can draw, but I'm not sure how to make that concrete.

Here's what I got so far,

  • Explain the conjecture, provide some simple examples to begin
  • Visual worksheet to get a 'feel' for it, (eg. Filling in blanks in on a diagram similar to the one on Wikipedia)
  • (For secondary students, having not mentioned whether its proved or not), practice forming an induction argument. Why does it fail? (eg. this post.)
  • Talk a bit about the twin prime conjecture and how it's related
  • (For any age) discuss the difference between showing the conjecture is true for many numbers, and a rigorous proof. (Eg. We know the pythagorean theorem is always true)

Are there any other interesting, mathematical concepts one can easily draw from the Goldbach conjecture?

Thanks!

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    $\begingroup$ Infinity is an important mathematical concept. You can quite easily prove that infinitely many even numbers are the sum of two primes, but that is not enough for the conjecture. $\endgroup$ – Dag Oskar Madsen Oct 26 '16 at 13:07
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    $\begingroup$ You should not just show all even numbers starting from 4 (it isn't true at 2, and 2 is even!) are sums of two primes up to some bound, but keep track of the number of ways this can be done, say with the first prime less than or equal to the second. The point is that the conjecture looks good numerically not because the numbers happen to be a sum of two primes at least once, but that the number of ways of doing it appears to be steadily growing. $\endgroup$ – KCd Nov 3 '16 at 22:16
  • $\begingroup$ Does this Youtube video help with the number line idea? I saw it several years ago and thought it was interesting, I had never thought about the Goldbach Conjecture in such a way. $\endgroup$ – ruferd Jun 27 '18 at 12:16
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I like to ask my probability students the question:

If you pick an integer between 2 and 100 uniformly at random, what's the probability that it's the average of two (not necessarily different) primes?

  • I like that above question (easily equivalent to Goldbach) because there's no preference for even versus odd numbers as in Goldbach. It also gets students checking a lot of cases. On the one hand, they quickly notice the pattern and make the conjecture themselves, and they also want to argue that the pattern holds for all numbers (that way they don't have to check 100 cases). But the beauty of it is that the prime numbers that show up feel random, and it very much feels like the pattern keeps working by accident. It's also a great opportunity to stress the difference between "seems to be true" and "it's always true." You could strip the probability from it if you like.

  • You could ask why the conjecture isn't about odd numbers. What's special about evens versus odds (there aren't that many even primes)?

  • Asking "are there infinitely many solutions to prime = prime + prime?" is equivalent to the twin prime conjecture and sounds Goldbach-like.

  • The whole conjecture is obviously false unless there are infinitely many primes. You could discuss that connection and why there are infinitely many primes.

  • You could say the weak Goldbach-conjecture "every number (greater than 1) is the sum of at most 3 primes." You could say why this is implied by regular Goldbach, and you could try to discuss why this doesn't immediately imply regular Goldbach (though that's tricky to do since both statements are likely true. And you can't really argue that two true statements don't imply each other.)

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Too top down and too proofy/hard in places. Do something more experiential and experimental.

"Explain the conjecture, provide some simple examples to begin"

Nope, run an exercise and have students find if they can figure out the sums (small numbers). Even let group pick a couple small numbers (maybe two digit ones) and let group find sum that works.

AT THAT POINT, state the conjecture (not as a famous conjecture, but just the concept) and just let class have a little debate on their own "if we had more time, do we think we could find a counterexample".

AFTER class debate...have them vote as a group (true/not true).

Then wrap after the vote, by giving the name Goldbach, saying when he lived and how long the conjecture has been around...computers checked numbers up to a gazillion (fill in number) and no counterexample found. BUT nobody can prove the concept (yet) either. What to do, what to do? (Class discussion, let them debate if conjectures like this should just be accepted or not...let them as a group debate the practical versus rigorous slant.)

THEN tell the story of FLT and how world had given up for centuries of ever finding a proof...and some nut in an attic for 7 years figured it out. [Play a little clip of Andrew crying or the like from the documentary...it's fair use to steal a snippet.]

"(For secondary students, having not mentioned whether its proved or not), practice forming an induction argument. Why does it fail? (eg. this post.)"

No. Too hard. Too boring. EVen for those with good algebra 2 background and having had induction method. Plus it doesn't get to a proof.

"Talk a bit about the twin prime conjecture and how it's related." OK.

"(For any age) discuss the difference between showing the conjecture is true for many numbers, and a rigorous proof. (Eg. We know the pythagorean theorem is always true)"

OK, but bring in earlier. And don't TELL. Have class debate the concept.

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One idea not mentioned already is to have students write a computer program to start checking every even number and see how quickly the code runs. This also requires that the program be able to generate prime numbers, so it's not completely obvious to write. Are there any tricks that can make the algorithm run faster? And how does the program slow down over time? If you graph n vs time to find n = p1 + p2, do you get a logarithmic curve? Lots to explore here.

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