Just how many prime numbers are there?

Over the past few months, my 4th and 5th graders had a number of opportunities to explore prime numbers and factoring – they drew factor trees, discovered divisibility rules, played Prime Climb, and more.  Last Sunday, I decided that they were ready to tackle the question of just how many prime numbers are there and the very related question, “is there a biggest prime number?”

Their opinions varied – I received a “yes”, a “no”, a “probably”, and a “yes and no.”  In fact, the kids had a great discussion on the subject.  It went something like this:

Kid A: There can’t be a biggest prime because there is no biggest number.
Kid B: No, that doesn’t follow because you can make infinitely many numbers with just a few primes.
Kid A: Oh yeah, you can use each prime any number of times in the factorization.
Kid C: So then we can’t tell whether there are infinitely many primes or not.
Kid B: I guess not.
Kid A: Yeah, I don’t know what I think any more.

Great, I definitely had them intrigued at this point. I told them that we were going to investigate the situation. We all agreed that if there are only finitely many primes, then there must be a biggest prime. And so, like Euclid over 2000 years ago, we supposed that this was the case and we named the prime p.

Next, I told the kids we were going to multiply all the numbers up to p and add 1 to the product (this step probably came out of nowhere for the kids). We called the resulting number n.

It took us several more minutes of discussion to agree that n is not divisible by any prime number less than or equal to p. And I was so glad when one kid exclaimed “But that doesn’t mean that n is prime!”

Indeed, but all of our earlier playing with primes had taught the kids that every whole number can be factored into primes. Thus, even if n is not prime, it has a prime factor and we had just agreed that it has to be greater than p.

At this point, the kids could sense that something was not quite right but they couldn’t quite articulate what. This was their first encounter with a proof by contradiction, so they were a little confused. How can it be that p is the biggest prime and at the same time n has a prime factor bigger than p?

And that was of course the point, that it can’t possibly be. But what did that mean? Luckily, we had done some logic with the kids earlier in the year, and so after some careful consideration we were able to conclude that this meant that there must not be a largest prime.

For a few moments after we made this conclusion the kids sat there very quietly (which is extremely unusual for this group!) and stared at me.  I finally broke the silence by saying “That’s pretty cool, right?”  I received a few yeahs and nods in reply, after which the kids were back to their usual, talkative and rowdy  selves.  But for 15 minutes that day, I had their full attention, they were interested, engaged, and they were thinking.  To quote a line from one of my favorite Russian movies, “I wouldn’t call it an act of heroism, but there’s something heroic about it.”


About aofradkin

I enjoy thinking about presenting mathematical concepts to young children in exciting and engaging ways.
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3 Responses to Just how many prime numbers are there?

  1. Joshua says:

    This could be a good follow-up investigation: another way to get primes.


    • aofradkin says:

      Definitely! I actually follow his blog and saw this post a few days ago. Very approachable and cool proof that I had not seen before. Thanks for drawing my attention to it again!


  2. David says:

    You could also look at it as a direct proof: You can use the trick to keep making more primes over and over, so you get infinitely many. I don’t suppose you want to start discussing the density of primes, eh? 🙂


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