Pi – 1 and some updates on own works

Only notes, when talking about me. Just thoughts. Will try to keep things short, precise, with much own insight.

By Euler’s sine infinite product for infinitesimal calculus, we have

<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />
\prod_{n=1}^{\infty} \left(\frac{2n}{2n-1} \cdot \frac{2n}{2n+1}\right) = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots = \frac{\pi}{2}.<br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />

that can be interpreted as take an even number, square it, and divide by multiplication of both previous and next numbers (odd, potentiall prime). do it for all numbers >2. More about it in G.C. Carr’s “Pure Mathematics”.

\frac{\pi}{2} = \prod^\infty_{k=1} \frac{(2k)^2}{(2k)^2-1} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots\ = \frac{4}{3} \cdot \frac{16}{15} \cdot \frac{36}{35} \cdot \frac{64}{63} \cdots\!

The product can be rewritten as which means take squares on even numbers and divide them by previous numbers.. and multiply till infinity. This one is better as shows clear connection between pi and squares (or potentially) dimensions of numbers (defined as u-function, earlier in the blog).

Staying close to square numbers, we have known solution to Basel problem (Euler), and again we find pi. Again, as always, a transcendental one.

Combining two former observations with, for even n, sin(n\varphi)=2^{n-1}sin(\varphi)cos(\varphi)(sin^2({{\pi}\over{n}}-sin^2(\varphi))(sin^2({{2\pi}\over{n}}-sin^2(\varphi))(sin^2({{3\pi}\over{n}}-sin^2(\varphi)).. that results in n=2^{n-1} sin({{\pi}\over{n}}) sin({{2\pi}\over{n}}) sin({{3\pi}\over{n}}).., we catch a cursory observation that \pi is embroidered in every geometrical mystery. Indeed, we interpret \pi in the context of arcs, circles, angles but rarely think of it in terms of length. Indeed, if we inclusion-exclusion (using odd division only) on a line of length 1, ie. 1-{{1}\over{3}}+{{1}\over{5}}-{{1}\over{7}}+{{1}\over{9}}={{\pi}\over{2}}, ie. again a number related to \pi, but this time in terms of length, with its interesting interpretation.

Combining those with one of the transpositions of prime numbers into  tensor (in that case only three dimensions, but pottentialy extensible, thus referred to as tensor-), we’d have an interesting idea for further research.

A couple of updates. Firstly, I’ve began to write a screenplay with my friend.

Secondly, I’ve noticed (me and my Friend) 24|(p_{k+1}^2-p_{k}^2), for primes 5,7,.. which is, though, a known result.

Thirdly, have analyzed this transformation, for the sake of the new approach to the FLT. Hope that this function looks interesting.

Besides those, I’ve come up (together with my close Friend) with an interesting approach to prime sieving based on {{n(n+p_k)}\over{6}} and similar (infinite number of such). Until I don’t have more quality information, I will not be posting much in this area.

Last but not least, poems (version one, not corrected much), https://mishabucko.files.wordpress.com/2012/02/disquisitiones_severus2.pdf

You can find it in the previous post (the link itself).

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About misha

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