When Does the First Film "Planet of the Apes" (1968) Take Place?

"Planet of the Apes" is a science fiction film written by Rod Serling with Michael Wilson [adapting "La Planète des singes" (1963) by Pierre Boulle] and released April 3rd, 1968. 

It takes place on the Planet of the Apes universe timeline in the year: 

3978 AD 

We know this because just before the three surviving astronauts disembark from the sinking spaceship in the second scene of the film, astronaut George Taylor manages a quick look at the ship's dashboard counter. It is stopped at 3978 AD. 

This counter is moving steadily forward in the first scene of the film, telling the American astronauts what year it is back on Earth, as they travel through outer space faster than the speed of light. (They have travelled from Earth in 1972 AD but, during their trip, apparently over 2,000 years pass on Earth because of time dilation due to time being relative based on speed because of gravity, as described by Einstein in the Theory of Relativity.) 

In the fifth film, "Battle for the Planet of the Apes" (1973), all these events are referred to as happening in 3950 AD. 

Although it is chronologically the second-to-last film in the original "Planet of the Apes" pentalogy, it can be watched first or entirely separately.