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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. Expressions of the form \eqref{1} are called Machin-like formulas.

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 $$\frac{\pi}{4} = \arctan \left( \frac12 \right) + \arctan \left( \frac15 \right) + \arctan \left( \frac18 \right) . \tag{3}\label{3}$$ 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 $$ \frac{\pi}{4} = 44 \arctan\left(\frac{1}{57}\right) + 7 \arctan\left(\frac{1}{239}\right) - 12 \arctan\left(\frac{1}{682}\right) + 24\arctan\left(\frac{1}{12943}\right) .\tag{4}\label{4}$$

(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?
  3. Do questions (1) and (2) become more palatable when we require $c_{k} = 1$ for all $k$ -- as in equation \eqref{3} ? If so, what are upper/lower bounds and the asymptotic growth rate in this case?

Note: I've also asked a version of this question on MSE.

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  • $\begingroup$ The table at the end of the linked page mathworld.wolfram.com/Machin-LikeFormulas.html gives lower bounds for $n\leq21$. $\endgroup$
    – Max Alekseyev
    Commented Apr 24 at 11:21
  • $\begingroup$ @MaxAlekseyev Yes, true. I wasn't sure though whether the table lists all known solutions, or just the ones found and listed by Weisstein $\endgroup$
    – Max Lonysa Muller
    Commented Apr 24 at 14:05
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    $\begingroup$ Just above the table there is an explanation what kind of solutions it accounts for. If there is no restriction for the kind of solutions, then we'll have $f(n+1)\geq f(n)$ thanks to formulas like (29) on the same page. $\endgroup$
    – Max Alekseyev
    Commented Apr 24 at 14:19
  • $\begingroup$ Are you aware of the connection of Machin-like formulas with gaussian integers ? $\endgroup$
    – Jean Marie Becker
    Commented Apr 25 at 7:06
  • $\begingroup$ @JeanMarieBecker Yes, but I wouldn't know how to leverage this connection to obtain bounds and asymptotics for $f(\cdot)$ $\endgroup$
    – Max Lonysa Muller
    Commented Apr 25 at 8:33

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