A function F is such that this relationship holds for all real x.

F(x) = F(398-x) = F(2158-x) = F(3214-x)

What is the maximum number of distinct values that can appear in the list F(0), F(1), F(2), ..., F(999).

Why yes, of course, further improvement is possible.

I have previously shown that the function values for the integers repeat with a cycle of 352, so the only values that need to be considered are f(1), f(2) ... f(352)

But we know that

f(352) = f(46)

f(351) = f(47)

.

.

.

f(200) = f(198)

f(199) = f(199)

So values f(200) through f(352) are not unique, and there are 153 of those.

Maximum distinct values = 352 - 153 = 199.

Further reduction might be possible, in that there might be some duplication among f(1), f(2), ...f(199). I will return to this later.