Lenstra->Bailey-> Triangle->pidgeon (ant) movement analysis, pi

From Lenstra(s) and Lovasz (LLL), till PSLQ in ten years (with improvements, almost two decades). One of the remarkable results of the algorithm is Bailey-Borwein-Plouffe (BBP formula), ie.

 \pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} \right) \right].

Furthermore, all those combined with polylogarithms (using the ladder mechanism), allowed implementation of certain improvements to the original formulas. Lets now notice the connection of the form of the equation to the Triangle I had created some time ago. From 8k+1, we’d have (counting the first child nodes) 8k+9, 92k+17, 1564k+109, etc. The convolution of reciprocals leading to this serendipity, ie. noticable similarity to the Triangle, makes me think about the following.

Take Langton’s ant into consideration.This 2-dim Turing machine, where ant is capable of universal computation (source: Wikipedia), with its unbounded trajectory (Cohen-Kung), makes the space covered with movement (steps) gradually increase, where the increase (after ca. 10k steps) suffers from a known phonomenon (source:Wolfram). Would there exist a bijective transformation of the lattice configuration into prime key sequence, then we’d have an opportunity to refer the movement itself (of the very ant) to an element (or configuration) of the Triangle, thus we’d be able to find the relations of the movement to the primes and pi. But, as we already know the infinite connection between primes and pi, we’d have just a piece of the picture to learn from.

Still, we know no connection between the ant’s trajectory and the primes. Therefore, it might be a better idea to come up with a new ant that moves accordingly to the primes (in some way). I have one, will present it.


About misha

Imagine a story that one can't believe. Hi. Life changes here. Small things only.
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