Integer solutions for exponential equation

[rahidz2003]

Problem : Solve the equation $\displaystyle 2^n+7 = x^2$, where n and x are integers. Explain why there are no other solutions.

OK, I have no clue how to crack this one. Obviously n=1, x=3 and n=1, x=-3 are solutions, but I don't know how to show that they're the only ones (are they?)

I did find that when n is 4, 8, 12, 16, etc, $\displaystyle 2^n+7$ ends in 3, so it can't be a square. And n can't be negative (because x^2 would be between 7 and 8), but that's all I can think of. Any ideas? Thanks!

 

[markbrock30]

I suggest you plot a graph in Excel for the function, that will show you why there is only one solution. The function on the left side of the equation rises steeply upwards after the intercept, and the other side of the equation grows out moves off to the sides of the graph more and more as time increases.

This means they will not intercept again.


Mark.