The Quest

I am an ant (or a number theorist) looking for the greatest happiness in small things.

I am king (or queen) of a great kingdom looking to expand my empire so that I
may find the greatest happiness wherever it may be.

I am an astronomer looking the farthest the eye and machine can see
into the depths of the heavens for it as well.

Hmm. I’m just a simple monk on top of an anthill in an orchard of a great kingdom beneath the heavens on a full moon night.

And I am enjoying the eternal bliss from within. Amen!

Dave.

My Sunny Star

(Moi)

You’re the one for me.

You bathe me in warmth and glow.

You smother me like no other star.

Even in my darkest hour,

You still have that ubiquitous afterglow

That I know.

It keeps me always near my sunny star.

(Star)

Trapped by my beauty,

I cast it away.

Trapped by my intensity,

I cast it away.

Away, away, away, …, I go

Until I am no longer

The light of your day nor your fading afterglow.

Dave.

A Brief Analysis of the Collatz Conjecture

Algorithm for the Collatz Conjecture.

In short the Collatz problem is simple enough that anyone can understand it, and yet relates not just to number theory (as described in other answers) but to issues of decidability, chaos, and the foundations of mathematics and of computation. That’s about as good as it gets for a problem even a small child can understand.” Matt.

Remark: If n = 1, the algorithm terminates all computations.

Crucial Idea: R+ is a dense set.

Remark: R+ is the set of all positive real numbers.

We claim

  1. n_{t} = (\frac{3}{2} )^{\frac{t}{2}} *  (\frac{3}{4} )^{\frac{t}{4}} *  (\frac{3}{8} ) ^{\frac{t}{8}} * ... *( \frac{3}{2^{k}} ) ^{\frac{t}{2^{k}}} * r = 1

over the Collatz sequence of positive odd integers, from n_{0} to n_{t} = 1 where the index, t, is the number of trials it takes the Collatz sequence of odd integers to converge to one.

Remark: There are no infinite (nontrivial) cycles of any length ( (\frac{3}{2} )^{\frac{t}{2}} * (\frac{3}{4} )^{\frac{t}{4}} * (\frac{3}{8} ) ^{\frac{t}{8}} * ... *( \frac{3}{2^{k}} ) ^{\frac{t}{2^{k}}} \rightarrow 0 as t \rightarrow \infty) in the Collatz sequence since the index, t, is clearly finite.

In addition, our claim is fundamentally based on the distribution of maximum divisors, 2^{i_{max}}, for the set of all positive even integers…

The positive real number, r, is determined by the algorithm for the Collatz conjecture. And therefore, its computation is generally complicated.

The variable, k, is determined by the maximum positive even number, e_{max}, in the Collatz sequence: k = \left \lfloor{\frac{log(e_{max })}{log(2)}}\right \rfloor.

Can we refute our analysis?

We assume maximum divisors, 2^{i}, for any positive even integer.

For any appropriate positive odd integer, n_{0} > 2^{68} -1, we have

n_{2} =  \frac{3n_{0} +1}{2^{i_{1}}} = \frac{3}{2^{i_{1}}} * r_{2} where r_{2} = n_{0} +\frac{1}{3}.

n_{3} = \frac{3(\frac{3n_{0} +1}{2^{i_{1}}}) + 1}{2^{i_{2}}} = \frac{3}{2^{i_{1}}} *\frac{3}{2^{i_{2}}} * r_{3}

where r_{3} = \frac{1}{9}(2^{i_{1}} + 9n_{0} +3).

n_{4} = \frac{3(\frac{3(\frac{3n_{0} +1}{2^{i_{1}}}) + 1}{2^{i_{2}}}) + 1}{2^{i_{3}}} = \frac{3}{2^{i_{1}}} *\frac{3}{2^{i_{2}}} *\frac{3}{2^{i_{3}}} *r_{4}

where r_{4} = \frac{2^{i_{1}}}{27}*(2^{i_{2}} + 3) + n_{0} +\frac{1}{3}.

Remarks: The values, 2^{i_{j}}, are distributed according to equation one:

n_{t} = (\frac{3}{2} )^{\frac{t}{2}} * (\frac{3}{4} )^{\frac{t}{4}} * (\frac{3}{8} ) ^{\frac{t}{8}} * ... *( \frac{3}{2^{k}} ) ^{\frac{t}{2^{k}}} * r = 1.

Moreover, r = r(n_{0}) is a positive real number.

Example: If we let n_{0} = 57, then we compute e_{max} = 196, k = 7, and t = 10.

Therefore, r = r(57) = 1/.0841394 = 11.8850384.

Dave’s Conjecture: r = r(n_{0 }, t) = O(t) or r = c_{t} * t for some real number, c_{t}, such that either c_{t}  >  1 or 0 < c_{t} < 1.

Example: If we have t = 1 for some n_{0 }, then as k \rightarrow \infty,

r = \frac{1}{ \prod_{i=1}^{k }(\frac{3}{2^{i}})^{\frac{t}{2^{i}}}} \rightarrow \frac{4}{3 } = c_{1 }.

Therefore, r = r(n_{0 }, 1) \approx \frac{4}{3} for infinitely many positive odd integers, n_{0} > 1.

Questions: What are the values for, c_{2}, c_{3}, c_{4}, ...?

Relevant Reference Links:

Wolfram Alpha Computation (r4);

Our_Response_to_4.2_A_probabilistic_heuristic;

Proof of Collatz Conjecture;

What is the importance of the Collatz conjecture?

The Collatz Equation that supports the Collatz Conjecture.

“Counting and ordering stuff (objects, sets, numbers, spaces, etc.) are fundamental.”

Where is peace found?

Home is where the heart is.

And where there’s a good heart, there’s a good home.

So peace begins and ends with you.

The world is a creation of your mind.

Never feel despair or feel helpless to change it.

It’s the change in you that matters most.

When you change, the world changes.

All the acts of the world are yours alone.

Yes, you can forgive yourself.

Yes, you can be at peace despite yourself.

And wherever you go, there you are.

You own it.

Peace eternal. Amen!

Backup: Get to know Maxwell’s Equations — KaiserScience

This is a backup of an article on Wired,’Get to know Maxwell’s Equations – You’re Using Them Right Now,” by Rhett Allain , 8/6/19 Maxwell’s equations are sort of a big deal in physics. They’re how we can model an electromagnetic wave—also known as light. Oh, it’s also how most electric generators work and even […]

Backup: Get to know Maxwell’s Equations — KaiserScience

We (mathematicians) have a valid proof of the Riemann Hypothesis that we can celebrate on the great Bernhard Riemann’s 194th birthday, September 17, 2020.

Gauss, Riemann, Hardy, and Hilbert would have been proud of us. Amen!

The central ideas behind the proof of the Riemann Hypothesis are deceptively simple. The proof depends on the very important Harmonic Series and the following important fact.

For every positive integer, n > 1, there exists a prime number, p, that divides n such that either p \leq   n^{\frac{1}{2}} or p = n.

Those two key ideas lead eventually to a valid proof of the Riemann Hypothesis. However, the final proof of the Riemann Hypothesis requires some hard work and some important results of Gauss, Riemann, Hardy, and others. 🙂

Relevant Reference Links:

https://www.researchgate.net/publication/300068776_Proof_of_Riemann_Hypothesis;

Proof of Riemann Hypothesis;

Why is the RH optimum?

Randomness can be a useful tool for solving problems.

On the Importance of the Periodic Table of the Elements:

Nature is great at making mass. Just study the periodic table of the elements, and we can see why. With such a wide diversity of elements and the many complex and diverse interactions, nature produces life as well. What a wonder!

And please continue to ponder our favorite equation, Work = Mass + Energy (environmental energy).

Let’s explore the table. First up, is the number two element of the table. It is Helium. Why Helium?

There has been some false reporting that the world (the planet, Earth) could run out of helium within the next decade. That is not true according to the story, That Dire Helium Shortage? Vastly Inflated, https://www.wired.com/2016/06/dire-helium-shortage-vastly-inflated/.

Helium is a noble gas, and its phase at room temperature is gas. And it is commercially recovered from natural gas deposits, mostly from Texas, Oklahoma and Kansas. Helium gas is used to inflate blimps, scientific balloons and party balloons. It is used as an inert shield for arc welding, to pressurize the fuel tanks of liquid fueled rockets and in supersonic wind tunnels.” according to the paper, The Element Helium, https://education.jlab.org/itselemental/ele002.html#:~:text=Helium%20is%20commercially%20recovered%20from,rockets%20and%20in%20supersonic%20windtunnels.

So fill up those balloons with helium, and enjoy the party. 🙂

Before we move on to our next element, we want to be clear that one of our goals is to study/research natural or artificial processes and systems to discover how to best meet human needs of energy, technology, and the sustainability of of both human and natural habitats on Earth.

And thus far, our related posts on this site are preliminary and undeveloped. That will change for the better…

The precious metal gold (Au) is quite popular in the news for good reasons. There’s plenty of debt worldwide, and it’s growing daily. And the world is awashed in fiat currencies…

How does one produce efficiently large quantities of gold in the laboratory ? And how does one effectively mitigate/solve the problem of harmful radioactivity associated with the artificial production of gold?

Relevant Reference Link: https://wtamu.edu/~cbaird/sq/2014/05/02/can-gold-be-created-from-other-elements/.