Good activity for an Introduction to Proofs class

I just read this post at Futility Closet. (Spoiler: Don’t click the link unless you want to know the punchline.) Perhaps the result is well known, but I hadn’t seen it before.

The post made me think of a neat project for an “Introduction to proofs” class. I’ll have to save it for the next time I teach that class (Discrete Mathematics at my college).

Start with the first n prime numbers, p_1,\ldots, p_n. Divide them into two sets. Let A be the product of the primes in one set and let B be the product of the primes in the other set (the product is 1 if the set is empty). For example, if n=4 we could have {2,3,7} and {5}, so A=42 and B=5. What can we say about A+B and A-B? Form a conjecture and prove it.

I like the problem because I could imagine the students will go through a process of (1) I have no idea. (2) Aha! I’ve figured it out. How do I prove it? (3) Wait, I found a counterexample. (4) Ahhh! I see what’s going on, here’s the theorem, and here’s the proof. (I think they would come up with a different statement of the theorem than the one in the Futility Closet post.)


  1. Japheth Wood says:


    This seems more accessible for hand computation than Euclid’s proof of the infinitude of primes.

    Would you expect any of your students to have seen Euclid’s proof? And would you expect that they’ll be fine with paper and pencil, or will they need a calculator (or go directly to Wolfram Alpha)?

    1. We do cover Euclid’s proof in that class. So this activity would fit in well. I’d have to think about when/how to do it (in class, at home?). I would probably let them use a calculator or Wolfram|Alpha to compute the various products.

  2. Interesting post. If you can elaborate more it will more useful for students.

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