Prime Number Spirals

 

I invite readers to review prior posts on prime numbers and Pi:

Ulam Spiral

Twin Prime Sandwich and a Prime Number Property Rediscovered

Estimation of Pi

I recently viewed a very graphic video by Grant Sanderson who ties the above ideas together with a prime number spiral (of course, I first thought it would be another take on the Ulam spiral but I was pleasantly surprised that there was another prime number spiral) that appears in polar coordinates. I can't simplify the video by Grant Sanderson (3Blue1Brown) any further but here is a link for your viewing: 

Why do prime numbers make these spirals?

My favorite part of the video is when Grant Sanderson reminds us math fans that the more we learn about various aspects of mathematics, the more we see the various connections (for instance the links between polar coordinates, pi and prime numbers).

My Favorite Math Websites

This list will grow in time as I'm sure I'll get additional suggestions.

Numberphile has audio and video podcasts. Producer Brady Haran does a very good job in the audio podcasts as he and his guests have to be able to explain mathematical concepts without any visual help. There are many topics also done on YouTube which allow visual aides.

https://www.numberphile.com/podcast/

Desmos is an on-line graphing calculator that allows one to explore relationships.

https://www.desmos.com/

3blue1brown, or 3b1b for those who prefer less of a tongue-twister, centers around presenting math with a visuals-first approach. Rather than first deciding on a lesson then putting illustrations to it for the sake of having a video, almost all projects start with a particular visualization, with the narrative and storyline then revolving around this image. 3b1b is hosted by Grant Sanderson.

https://www.3blue1brown.com/

Favorite place to explore integer sequences: On-Line Encyclopedia of Integer Sequences (OEIS): https://oeis.org/

Favorite Math comedian: Matt Parker (a frequent contributor to Numberphile)

Matt Parker | Standup Mathematician (standupmaths.com)

Favorite Math author: Alex Bellos
Alex Bellos

Best explanation of math without numbers:
https://milobeckman.com/home-2021.html
Also see post: https://jamesmacmath.blogspot.com/2021/03/math-without-numbers-book-review.htmlMath Without Numbers

Best middle-school to high-school resource - Eddie Woo's: Wootube – Find joy in learning mathematics. (misterwootube.com)

Eddie Woo also authored:
It's a Numberful World: How Math is Hiding Everywhere (Books + TV – Wootube (misterwootube.com)

I've also become a fan of Jordan Ellenberg, author of 

Shape, The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else 

and

How Not To Be Wrong, The Power Of Mathematical Thinking

Favorite Independent Science: A New Kind of Science

A New Kind of Science - Wikipedia

A great deal - free e-book. This is a magnificent way to share knowledge and encourage exploration of a completely different way of approaching math and science.
Stephen Wolfram: A New Kind of Science | Online—Table of Contents (wolframscience.com)

Update 7-28-2021 An article about Wolfram's and other scientists' effort to explain why the universe exists: Why does the Universe exist at all? - BBC Science Focus Magazine

Brussels Choice

This post is inspired by my friends who are driven by the Collatz Conjecture (see prior posts: HOTPO, additional thoughts). The  Numberphile Podcast has a short video introducing the Brussels Choice, a problem of sequences. The guest on this episode is the founder of the OEIS (the On-Line Encyclopedia of Integer Sequences), Neil Sloane.

Like the Collatz Conjecture, one starts with any integer and follows simple rules to convert the number to 1 (or other numbers) in a series of steps. Unlike the Collatz Conjecture, not all numbers can be converted to 1. Numbers ending in 5 and 0 cannot be converted to 1 but can be converted to 5.

Rules:

• Any digit or sequence of digits within the number that ends with an even number can be doubled or halved. The other digits are unchanged.
• Any digit or sequence of digits within the number that ends even an odd number can only be doubled. The other digits are unchanged.

Example - start with 6113

Double the 3, the remaining digits are unchanged: 6116

Divide 16 by 2: 618

Divide 8 by 2: 614

Divide 4 by 2: 612

Divide 2 by 2: 611

Divide 6 by 2: 311

Double the final 1: 312

Divide 312 by 2: 156

Double 1: 256 (a power of 2)

Divide 256 by 2 and repeat 7 more times to reach 1.

Example - start with 90 (numbers ending with a 0 or 5 can be converted to 5)

Double 9: 180

Divide 8 by 2: 140

Double 14: 280

Double 28: 560

Double 56: 1120

Divide 12 by 2: 160

Divide 16 by 2: 80

Divide 8 by 2: 40

Divide 4 by 2: 20

Divide 2 by 2: 10

Divide 10 by 2: 5

The site Code Golf, has a challenge to write the code to determine if two numbers are connected by the Brussels Choice.

What is the next number in the sequence...?

In 1964 Neil Sloane started the maintenance of a list of integer sequences. Once his collection grew, he published the list in a book in 1973 ("A Handbook of Integer Sequences", by NJAS, Academic Press, NY). This book contained 2372 sequences.

Since 1996. the list in the form of a database has been maintained on the internet. As of the date of this post, it has over 300,000 different sequences that can be searched in many different ways. The formal name is The On-Line Encyclopedia of Integer Sequences or OEIS. This is a link to the database.

The database is copyrighted by the OEIS Foundation, Inc. 

Just to try out, I entered "1,2,3,4,5" in the search function. As expected, the first sequence listed was the positive integers (OEIS sequence identified as A000027). What I didn't expect was that there were 6820 other sequences that begin "1,2,3,4,5."

Many famous sequences are included such as the list of prime numbers, A00040, the Fibonacci numbers, A000045, and Pascal's triangle, A007318.

3-16-2021 Update.

I suggest viewing Mathologer YouTube entry on "What Comes Next?"

Also, view these Numberphile videos showing some of these sequences graphed:

Amazing Graphs

Amazing Graphs II

Amazing Graphs III

3-26-2023 Update: My proposed sequence was published in the OEIS - A361746 - OEIS.

To view any of the sequences that I've authored or to which I have contributed, see: james c. mcmahon - OEIS.

Tribute to Ronald Graham's Largest Number ...262464195387

What is the largest number? Any number that is offered can be bettered by that number plus 1. A common answer is infinity, although infinity isn't a specific number. "Infinite" describes something that is without bounds. Something could be infinitely large (set of integers) or something could be into infinitesimally small (as done in calculus). 

Another approach to this question of the largest number is to ask what is the largest number used in a proof. In 1977 the mathematician Ronald Graham established the world record for the largest specific integer used in a mathematical proof. Graham's number is so large that if all the atoms in the universe were made into ink, the number could not be written. That is certainly disappointed for the readers of this post who wanted to see the number.

However, the last digits of Graham's number are: ...262464195387.

The actual expression of Graham's number uses hyperoperators which are higher order forms of exponentiation. 

Since 1977, Graham's number has been exceeded by larger super-numbers used in proofs. One example is TREE(3).

Ronald Graham passed away in 2020 and was honored in a Numberphile podcast. An earlier podcast specifically described Graham's number. While the end digits of Graham's number have been determined, the first digit is unknown. Graham was asked what digit he would like it to be and he said he actually knew the first digit but only when the number is expressed in base-2 and then it is 1.

The icon I chose for this post is juggling because Ron Graham was an accomplished juggler. For a photo of Graham, see New York Times Obituary Link with photo.

A Simple Pythagorean Theorem Proof

 

I recently came across a proof for the Pythagorean Theorem that is simple and contains a minimum amount of equations. It may be one of the most direct and understandable proofs of the many that exist. A nice animation of this proof is found at MathAdam.

 

Start with your standard right-angle triangle of sides A, B and C with C being the hypotenuse.

 

Draw a line perpendicular to C to the opposite vertex.

Now, in addition to the original triangle, there are two additional right-angle triangles formed. One has the side A as its hypotenuse (well name triangle A) and the other has side B has its hypotenuse (triangle B). The original triangle with sides ABC will now be called triangle C. The three triangles, A, B, and C are all similar A ~ B ~ C. (This can be established by the fact all three has the same interior angles.) Another key fact that we can see is the area of A + B =  area of C

The next step is to flip over each of the three triangles.

The next step is to the draw out squares on each of the three sides.

The relative sizes of the squares have the same relative ratios as the triangles A:B:C.

Since the areas of triangles has the relationship: A + B = C, then the same applies for the squares, therefore A² + B² = C².