More VW - hmm it's getting hard :P
by Wolstenholme, Nov 28, 2014, 5:09 AM
Well the identity basically gives away that it is in fact an algebraic integer.
We proceed with a lemma.Lemma 42:
is a unit in ![$ \mathbb{Z}[\omega]. $](http://latex.artofproblemsolving.com/a/0/5/a055366e32a8a733bd1bc9f178f77c66bb3dc47c.png)
Well it clearly suffices to show that
where 
denotes the 
-th cyclotomic polynomial. I shall show by induction that 
for all positive integers 
not equal to a power of a prime and 
if is a power of the prime 
The base case is trivial so using the identity 
and dividing both sides by 
immediately yields the desired result.By Lemma 42 it suffices to find
such that 
is an algebraic integer. Now it is clear that is an algebraic integer if and only if 
is an integer where 
and 
or if one of 
and 
is divisible by 
Using the results from the proof of Lemma 42 we get an answer of
August 2016
June 2016