To solve the problem, imagine you are on the peak of the mountain that you want to be able to see from far away. Now imagine a line from your eye that goes all the way to towards the horizon, where it becomes a tangent to Earth's curvature. The distance from your eye to the point of tangency is the visible distance.
For all practical purposes, because Earth's radius is so big (3,960 miles), you can ignore how high above sea level you are and that Earth's radius varies a little (+/- 20 miles).
The tangent is perpendicular the Earth's radius, therefore the Pythagorean theorem allows us to calculate the visible distance (distance between point of tangency and eye). BITOG won't let me type the square root sign, so I use ^(1/2) instead:
(radius + height of mountain) ^2 = radius ^2 + tangent ^2
tangent ^2 = 2 x radius x height + height ^2
tangent = ^(1/2) 2 x radius x ^(1/2) 1+ height ^2 : 2 x radius x height
tangent = ^(1/2) 2 x radius x height (1 + height : 4 x radius
tangent = ^(1/2) 2 x radius x height
The visible distance D (from point of tangency to eye) is the square root of two times Earth's radius times height of the mountain:
D = ^(1/2) 2 x 3,960 x height of mountain in miles
For example, if a particular mountain peak is 10,560 feet (2 miles), then:
D = ^(1/2) 7380 x 2
D = 121
You can see a two miles high peak from about 121 miles away, presuming you are reasonably close to sea level (a few hundred feet don't matter) with a clear view. Due to refraction you can actually see farther, if atmospheric condition allow it.