Who Am I?
I am a proper fraction.
The sum of my numerator and my denominator is 1 less than a perfect square.
The difference of my denominator and my numerator is 1 more than a perfect square.
The product of my numerator and my denominator is 1 less than a perfect square.
What fraction am I?
Hint: First, try some examples of proper fractions, so that you get a handle on the problem.
Now, focus on the nature of the possible squares involved...as they contribute some great information.
Solution Commentary: One solution is 3/5....but there are others!
So,to guide your investigation, suppose A/B is your desired fraction, where A+B = C^{2}1 and BA = D^{2}+1 and AB = E^{2}1. Now, here are some more important questions to try to resolve in order to generalize the problem:
Can A and B both be even? Both be odd? One be odd and one be even?
Which is greater, C or D? Can you tell anything about the value of E relative to the values of C or D?
Can C and D both be even? Both be odd? One be odd and one be even?
Can C = D?
Is it necessary that C  D = 2?
Since AB = E^{2}1, does it help to rewrite this expression as AB = (E+1)(E1)?
Once you find some solutions, can you find a pattern in these solutions?
Have fun.......and what happens if the fourth clue was changed to "The product of my numerator and my denominator is 1 more than a perfect square"?
