Sunday, April 12, 2009

Number Theory, Logic, Proofs and Patterns for Middle School and Beyond...


The following is a series of apparently straightforward arithmetic problems for middle schoolers. However, the objective is to have students justify their reasoning beyond "guess and test" methods. Proving there is only one solution or none requires more careful logic using algebra as needed. Students will need some basic algebra for the "proofs." For the younger student, modify these questions to have them find the squares in questions 1,2,3 and 5. Take this as far as you wish...

In the following, square refers to the square of an integer. Justify your reasoning or prove each of the following.

(1) There is only one square which is 1 more than a prime.

(2) There is only one square which is 4 more than a prime.

(3) There is only one square which is 9 more than a prime.

(4) There is no square which is 16 more than a prime.

(5) There is only one square which is 25 more than a prime.

(6) Can one generalize this or not??

Click Read More for selected answers, solutions...

Selected Answers, Solutions

(2) If n2 is 4 more than some prime, p, then we can write
p = n2 - 4 = (n-2)(n+2). Since p is prime, the smaller factor must be 1, so
n-2 = 1 or n = 3. Thus, there is only one square, 9, which is 4 more than the prime, 5.

(4) p = n2 - 16 = (n-4)(n+4). There n would have to equal 5, n2 would equal 25 but 25 - 16 = 9 is not prime.

(6) If there were a general rule would that mean we'd have a formula for primes?

Your thoughts about these questions...


Al said...

Nice problem. For middle school students this could a nice chance to not only have them find the values but then introduce them to the proof afterwards by modeling it for the first problem and having them replicate the method for the others.

Dave Marain said...

Thanks, Al...
I like your idea about how to present this to your students. I am particularly impressed by your willingness to try proof arguments with that age group using a model for them to follow.

I recognize that all of those questions are basically the same but I provided several examples for students to investigate an interesting relationship which might lead them to think primes might follow a pattern which of course they don't. Well, not any that us humans have yet discovered!

Joshua Zucker said...

There was an old ARML problem that asked for the smallest value of n such that n + p is never a square for any prime p, or something like that, with the answer being 16.

So another question in addition to yours here, is whether every non-square n has the property that n + p is eventually a square for some prime p.

Dave Marain said...

I miss those ARML challenges. Thank you for recalling that.

Using differences, we are really talking about finding integer solutions (or proving there is none) for the equation
n^2 - k = p, where p is prime.
If k is a square, then 16 is the least positive integer for which there is no solution. So we can now ask if there are other values of k.

If k is a non-square, like 3 or 15, then it is possible to find numerous primes of the form
n^2 - 3 or n^2 - 15.

Numbers of the form n^2 - k are usually termed "near-square numbers" and if they're prime, then "near-square primes". What the applications of all this are I'm not sure but it does make for an interesting investigation.

I believe that most mathematicians were attracted to the subject by their fascination with the curious properties of primes and there are numerous wonderful websites devoted to this. One of the best is known as The Prime Pages. I may have more to say later on.