Hello, perash!
I haven't solve it yet . . . It's quite involved . . .
The Captain is absolutely correct . . .Solve the following equation on the set of integers:
We have: .
We know that: . . . . for any integer
Hence: . must be an integer.
. . We can let: .
There are more restrictions:
. . must be nonnegative: .
. . And, of course, must be a square.
16 doesn't work.
I have "reduced" the problem to solving the Diophantine equation:
(I can explain how I got this, but I'd like the original poster to do some work on their own.)
So
There are going to be some obvious limits on m and n here because n + 2m has to be one of 1, 3, 19, 57.
So we are looking the solution to one of two systems of equations:
and
or
and
The first has a solution of m = 14 and n = 29, giving an x value of -13. The second has a solution of m = 4 and n = 11, giving an x value of 7.
So the only two solutions are x = -13 and x = 7.
-Dan