Abstract
We analyze a problem of interactions between elements of an ideal system which consists of two point masses and a wall in a hyperbolic setting. Thanks to a change of variables, the problem is reduced to a sequence of reflections on a hyperbola. For specific ratios of the two masses, the number of interactions is related to the first numerical digits of the logarithmic constant ln (2).
Highlights
In [1], the author proposed the computation of the first digits of π by counting the number of collisions of a system consisting of two balls and a wall
Thanks to a useful transformation, the basic geometry of the problem is reduced to reflections on a hyperbola
For two particular ratios of the masses, the analysis leads to the computation of the first digits of any logarithmic constant ln ( p) [3]
Summary
In [1], the author proposed the computation of the first digits of π by counting the number of collisions of a system consisting of two balls and a wall. Extensive analysis of this problem has been done, see [2] and references therein. Thanks to a useful transformation, the basic geometry of the problem is reduced to reflections on a hyperbola. It follows that a sequence of interactions becomes a sequence of reflections. For two particular ratios of the masses, the analysis leads to the computation of the first digits of any logarithmic constant ln ( p) (where p is a prime number) [3]
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