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...