Apparently my amazing "

proof" that all

coefficients of a cyclotomic polynomial are -1, 0 or 1 is flawed, since the

105th cyclotomic polynomial has this sequence of coefficients (generated using

GAP):

[ 1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, -1, 0, -1,
0, -1, 0, -1, 0, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, -1, -1, -2, -1, -1, 0,
0, 1, 1, 1 ],

which contains

-2.

MathWorld explains that this is due to 105 = 3*5*7 being a product of

3 odd primes (clearly it is the first such).

Instead, I give you the following trivial fact about cyclotomic polynomials. Their sequence of coefficients is a palindrome: it reads the same in both directions, possibly changing signs on all coefficients! This is less astounding when you think about it. Here's why the sequence of coefficients of a cyclotomic polynomial is a palindrome...