TITLE: A problem on sums of arctangents of rationals
QUESTION [5 upvotes]: Let $S$ be a set of rational numbers.
For "special" sets $S$, we can ask if $\pi$ can be written as a $\mathbb{Q}$-linear or $\mathbb{Z}$-linear combination of elements from '$\{\tan^{-1}(x): x \in S\}$'.
In particular, let $\hat S$ be the closure of $S$ under both negation and the binary operation $p*q=\frac{(p+q)}{(1-pq)}$.

For any natural number $b \geq 2$, $S_b =\{ \frac{1}{b^k}: k \geq 1 \}$. Prove that $0 \notin  \hat {S_b}$.
Note: $1 \notin \hat S$ since any rational number $P/Q$ in $\hat {S_b}$ in non-reduced form satisfies $(P,Q) \in \{(0,1), (0,-1),(1,0),(-1,0)\}$ (mod $b$).

$tan^{-1}(1/2) + tan^{-1}(1/3) = \pi/4$.
Let $b_1,b_2 \geq 2$ be two natural numbers. For which pairs does 1 belong to the closure of $S_{b_1} \cup S_{b_2}$? For which pairs does zero belong to the closure? Are there pairs for which $\pi$ can be written as a $\mathbb{Q}$-linear combination of the arctangents of their negative powers but not as $\mathbb{Z}$-linear combinations?

REPLY [3 votes]: In addition to Paul's comment let me point out that the structure of the underlying group was investigated in 

Sam Northshield, Associativity of the secant method, Amer. Math. Mon. 109 (2002), 246-257,

and that the relation with the arithmetic of ${\mathbb Z}[i]$ was observed by 

John Todd, A problem on arc tangent relations, Amer. Math. Mon. 56 (1949), 517-528

The associativity of the secant method is, by the way, a consequence of the usual geometric interpretation of the group law on conics.