(1) See the visualization for the difference of squares posted on 1-9-08.
(2) Read the comments in this post for considerable clarification and instructor guidelines and suggestions. Mathmom's and Eric's comments are particularly insightful.
This post can be developed into an activity for prealgebra through first-year algebra students (or even 2nd year algebra). The last part is more challenging.
The focus here is on developing a method/strategy that can be used to solve similar Diophantine equations. The other objective is to introduce the ideas and methods of proof. This problem may later be used to solve a recent math contest problem for which I obtained permission to discuss on this blog. I am fully aware that many students will 'solve' these equations by Guess-Test methods, but they need to go further.
(a) Prove there is only one solution in positive integers for the equation:
M2 - N2 = 12
Note: If we omit the word positive, what would the solution(s) be?
(b) Determine all positive integer solutions:
M2 - N2 = 15
(c) Determine all positive integer solutions:
M2 - N2 = 36
(d) Let's investigate for what positive integer values of P, M2 - N2 = P has NO solutions in positive integers.
(i) Determine at least 5 positive integer values of P for which the above equation has no positive integer solutions.
(ii) (More challenging) Describe all values of P for which the above equation has no solutions. Justify your result.
Note: All students should have success with (i), although some may struggle to find 5 values. Part(ii) should challenge the student who has finished the other parts in rapid order and sits there complacently!
Additional Comment: If P is itself a perfect square, our equation is obviously related to the most famous equation in geometry. Thus, if P = 9 or P = 16, for example, students should recognize something! For this reason you may want to have students consider these values when doing this investigation. More to come...
Tuesday, January 8, 2008