I am kind of stuck on a practice problem relating to proof by contradiction that goes as follows:
"Prove that there do not exist positive integers $m$ and $n$ such that $m^2 – n^2 = 1$"
For the outline of my proof (which is quite long and inefficient): I assumed, to the contrary, that there exists positive integers $m$ and $n$ such that $m^2 – n^2 = 1$"
Then considered four possible cases where:
case 1: $m$ even, and $n$ odd, and showing that $m^2 – n^2 = 1$ is a contradiction.
case 2: $m$ even, $n$ even, and showing that $m^2 – n^2 = 1$ is a contradiction.
case 3: $m$ odd, $n$ odd and showing that $m^2 – n^2 = 1$ is a contradiction.
case 4: $m$ odd, $n$ odd and showing that $m^2 – n^2 = 1$ is a contradiction (things got a little sticky here) so I used a lemma given by:
lemma: The product of two consecutive positive integer, cannot be the square of an integer.
Now where I ran into a problem was proving the lemma (which I am not 100% is true or not but intuition tells me that it seems like it), which basically renders my 4th case invalid.
I am pretty sure that I am doing unnecessary work here and would greatly appreciate feedback since I am trying to practice and the textbook didnt give a solution to this problem.
Thank you.
Best Answer
All you have to do is factor and assume the statement is true. We see that $$m^2 - n^2 = (m + n)(m - n) = 1.$$
If $m$ and $n$ are positive integers, both $m + n$ and $m - n$ must also be integers. Thus, we must have that $m + n = m - n = 1.$
But clearly, this is not possible - the only solution to the system is $m = 1$ and $n = 0,$ which is not a positive integer!
And we are finished! Hope this helped!