Upper and lower bounds on the number of solutions to the equation $\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}} \right) $

Question

Background

The Norwegian mathematician and astronomer Carl Størmer did important work on the equation

$$\frac{\pi}{4} = \sum_{k=1}^{n} c_{k} \arctan \left(\frac{1}{x_{k}}\right), \label{1}\tag{1} $$

where $c_{k} \in \mathbb{Z} \setminus \{0\} $ and $x_{k} \in \mathbb{N}_{> 0}$ for all $k$. Here, the numbers of the form $\arctan \left(\frac{1}{x_{k}}\right)$ are called the Gregory numbers.

For $n=2$, there is the original formula due to Machin: $$ \frac{\pi}{4} = 4 \arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right). \tag{2}\label{2} $$ Størmer found three additional solutions to \eqref{1}, and established there are no more than four solutions when $n=2$. He proceeded by looking for solutions in the case when $n=3$, and found 103 cases. An example of such a solution is $$\frac{\pi}{4} = \arctan \left( \frac12 \right) + \arctan \left( \frac15 \right) + \arctan \left( \frac18 \right) .$$ However, he couldn't show there are no more solutions.

As Nimbran describes in the following 2010 paper (PDF), two additional solutions were found by J. M. Wrench in 1938, and one more was found by Hwan Chien-lih in 1993. It appears to be an open problem whether these are all solutions.

For $n=4$, Størmer also obtained solutions. For instance, we have $$ \frac{\pi}{4} = 44 \arctan\left(\frac{1}{57}\right) + 7 \arctan\left(\frac{1}{239}\right) - 12 \arctan\left(\frac{1}{682}\right) + 24\arctan\left(\frac{1}{12943}\right) .\tag{3}\label{3}$$

(See p. 5 of the paper by Nimbran.) Again, the total number of solutions appears to be unknown.

Let $f(n)$ be the number of solutions to \eqref{1} for $n \geq 1$. We have:

$n$	$1$	$2$	$3$	$4$
$f(n)$	$1$	$4$	$\geq 106 $	?

Questions

1. Are there any upper and lower bounds for $f(n)$ ?
2. Can any results on the asymptotic growth rate of $f(\cdot)$ be established?

Answer 1

REMARKS.- $(1)$ You have to edit Machin's equality : which must be $$\frac{\pi}{4} = 4 \arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right) $$ (what you have written gives $\pi$ instead of $\dfrac{\pi}{4}$).

$(2)$ Similar to Størmer's equality with four addens in XIX century, we have Takano's equality (1982) which is $$\frac{\pi}{4} = 12 \arctan\left(\frac{1}{49}\right) + 32 \arctan\left(\frac{1}{57}\right) - 5 \arctan\left(\frac{1}{239}\right) + 12\arctan\left(\frac{1}{110443}\right)$$ $(3)$ Suppose $$\frac{\pi}{4}=\sum_{i=1}^{i=4}a_i\arctan\left(\frac{1}{b_i}\right)$$ of which one has the two examples of Størmer and Takano. We note that for each of these two examples, if there exist $i\ne j$ such that $$a_i\arctan\left(\frac{1}{b_i}\right)+x\arctan\left(\frac{1}{y}\right)=c\in\Bbb R\\a_j\arctan\left(\frac{1}{b_j}\right)+z\arctan\left(\frac{1}{w}\right)=-c$$ then we would have found the following example with six addends:

$$\frac{\pi}{4}=\sum_{i=1}^{i=4}a_i\arctan\left(\frac{1}{b_i}\right)+x\arctan\left(\frac{1}{y}\right)+z\arctan\left(\frac{1}{w}\right)$$