Problem 11 – February 1st, 2019
When Are There Integer Solutions?
Describe all positive integers a for which the equation x2 – y2 = a has integer solutions.
Submit your answers to mathpotw@acu.edu. Details for submissions can be found here.
Correct solutions were submitted by Wyatt Witemeyer.
Solution to When Are There Integer Solutions?
The expression x2 – y2 factors into the product (x – y)(x + y). Now x – y and x + y have the same parity, that is, they are either both even or both odd. If they are both even, then their product is a multiple of 4. If they are both odd, then their product is odd.
Thus the positive integer a must either be odd or be a multiple of 4.
