Over on https://mathlesstraveled.com there’s a series of posts going on having to do with this:
See that? As increases, approaches integer values. Odd, huh? Why does it do that?
Despite what should have been a dead giveaway hint, I didn’t figure out the approach revealed in this post. Embarrassing. But having failed on the insight front, what can I do on the obvious generalization front?
Let’s think about quantities of the form
where , , , , and are nonzero integers; is in lowest terms and , , and . For now let’s also restrict to primes.
To investigate that we’ll consider
The complex quantities lie on the unit circle in the complex plane and are the vertices of an -gon. Using the binomial expansion, the sum is
Now, for the terms where is a multiple of , is equal to 1 and the sum over equals .
Otherwise, we’re summing over the points on the unit circle:
which is the sum of a geometric series so
For instance, when , the sum is . When , it’s .
All right then. This means we keep only the terms where is a multiple of :
which is an integer. Call it . Then
So for large , approaches an integer if and only if the magnitudes of all the quantities have magnitude .
For instance: Let , . Then
Now in the case ,
The magnitude of for . In fact it’s zero for , but then is an integer anyway; point is, or works: and both approach integers (the former much more quickly than the latter).
How about ? Then
By symmetry the magnitudes of the two complex numbers are the same, so what we need is
So there are no integer values of that give convergence to an integer for . It seems evident the same is true for all .