## Decimal expansion of pi

Today, no docs, nothing special, a couple of updates on thinking.

Firstly, see: http://oeis.org/A005042 and http://oeis.org/A060421 for reference. Finally refer to http://oeis.org/A000796. How can we expect prodigy killer action from these?

If we were to find all next primes in the very expansion, that’d be more appealing, albeit still – it’s decimal approach. And we know (from experience, though) that multiplication measures applied to primes equal (currently) unsolved problems (eg. the Goldbach Conjecture).

Thus, I recently tried to approach all these from a different perspective. Firstly, it’s trivial to notice that the GC is equivalent to $p= {{p_1+p_2}\over{2}}-{{p_3+p_4}\over{2}}$. Assuming the correctness of the former (even Euler did that), we may notice (due to AM-GM) the geometrical connections between four different primes. This can be shown as $4p^2=(p_1-p_3)(p_1+p_3)+(p_2-p_4)(p_2+p_4)+2(p_1p_2-p_3p_4)$, ie. can be shown as sumations of areas of rectangles, which I will show later in my sketch.

But, omitting that observation, I noticed yet another relevant approach. Lets introduce $j(t)=\sum\limits_{p} (-1)^{i} \left \lfloor \dfrac {t} {p} \right \rfloor$, where $i \in N$ means $i$-th prime used. It’s straightforward that by inclusion-exclusion we have some sort of representation of the number of primes (because we sieve certain numbers). Considering the function, lets now strike out the floor function, thus we have $j(t)=\sum\limits_{p}{{t}\over{p}}(-1)^{i}$, (where $i$ indicates $i$-th prime number, its index) which pinpoints its potential connection to Zeta(2) and I-E representation of pi.

I forgot to post the two limits I have recently had trouble with regarding the introduced triangle:

$lim_{k\rightarrow\infty}({{2k+1}\over{2k+3}}+{{2k+3}\over{2k+5}}+{{2k+5}\over{6k+5}}+{{6k+5}\over{6k+7}}+{{6k+7}\over{10k+7}}+{{10k+7}\over{10k+9}}+{{10k+9}\over{30k+11}}+{{30k+11}\over{30k+17}}+{{30k+17}\over{30k+23}}+{{30k+23}\over{30k+29}}+{{30k+29}\over{42k+13}}+...)$

$lim_{k\rightarrow\infty}({{2k+1}\over{6k+5}}+{{6k+5}\over{30k+11}}+{{30k+11}\over{42k+13}}+...)$

Assuming the first could be divergent due to the amount of ones, I decided to introduce the second one. Waiting for solutions.

Besides all those, I had found a nice proof for (mine)

Prove that no $(x,p), x,p \in N, x\neq 1$, with $p$ prime, exist such that $x^3+3x^2(1-3p)-2p^2(12x-13p)=0$. I have a solution.

Also, as you know, I’m asking a question whether we can create a number with infinite number of divisors, thus contradicting the GC. But, due to our axioms, we can’t (explained by the loop: we can’t create as we could find yet another prime to multiply by). But, can we even imagine such a model where it’d be possible, remembering Russels paradox in the set theory and Cantor’s countability of infinities, ie. primary examples of axiom verification.

Last thing, will also be shown in the sketch later. In one of my visions, I recognized primes as regular geometrical objects, e.g. 5 would be regular pentagon, 7 would be regular heptagon etc. And I wondered how would it be possible to create such mosaics built of such objects? I’m sure I will write about this later, again in this notebook.