This is our 512th post

Jim Carrey is an actor known best for his comedic roles, and is considered one of the top movie stars in Hollywood. He starred in the movie The Number 23, which is based on the the 23 enigma: a belief that all of life is directly connected to the number 23, often in an indirect manner.

Today Ken and I want to thank everyone who reads and supports GLL; thank you very much.

Because this is our ${512^{th}}$ musing, we want it to be brilliant, unique, and fun. Well we have many things that we would like to talk about, but none seemed that special. After all the next power of two is distant, so this is our last chance for a long time to highlight the cardinality of the post.

We could talk about our secret polynomial time algorithm for factoring ${\dots}$ Oops. I forgot that is supposed to be kept secret. Actually, if the algorithm is a Leonid Levin-style stealing or adapting algorithm, then is the algorithm itself the secret? We don’t know, so instead we will talk about something else.

## The 23 Enigma

As I started to think about the significance of ${512}$, I realized the number itself is of supreme importance. Perhaps there is some truth in the 23 enigma that is the basis of Carrey’s movie—which by the way was panned by the critics. Carrey was nominated for the 2007 Golden Raspberry Award for Worst Actor, but lost out to Eddie Murphy in the movie “Norbit.” I am thankful that in our area of endeavor we only give positive awards. What would it be like to get the “worst published paper award of 2013”, or to lose by one in that category?

Back to the 23 enigma. My first observation was that David Hilbert’s famous problem list had twenty-three problems—hmmm. I started to look to see if there were any other connections. I next noted that

$\displaystyle \begin{array}{rcl} 23 \times 23 - 2^3 - 3^2 &=& 512;\\ 23 \times 22 + 2 \times 3 &=& 512. \end{array}$

I had originally thought that the more obvious way of making ${512}$ as ${(2^3)^3}$ didn’t balance out, but Ken pointed out that the outer part can still be read “to three” in English. Well, this is just chance—right? But how about these:

• “Dick Lipton & Ken Regan” has 23 characters, including spaces;
• “SAT in polynomial time?” has 23 characters;
• “Prof. Richard J. Lipton” has 23 characters;
• “Doctor Kenneth W. Regan” has 23 characters;
• “Does P equal NP or not?” has 23 characters;
• “P could equal NP really” has 23 characters;
• “The twenty-three enigma” has 23 characters;
• “Twenty-three characters” has 23 characters.

Now “Gödel’s Lost Letter and P=NP” has ${23 + 2 + 3}$ characters, but also its name suggests counting letters. It has 22 letters, but remember there’s a lost letter, so it’s originally 23. And if you’re thinking of a “factoring algorithm,” what name would you most likely put in front? Rabin or Levin, right? Or John Dixon, or someone with “Peter” in his name. All add up to 23 letters.

## The History of the 23 Enigma

According to our friends at Wikipedia: Robert Wilson cites William Burroughs as being the first person to believe in the 23 enigma. Wilson, in an article in the still-running magazine Fortean Times, related the following story:

I first heard of the 23 enigma from William Burroughs, author of Naked Lunch, Nova Express, etc. According to Burroughs, he had known a certain Captain Clark, around 1960 in Tangier, who once bragged that he had been sailing 23 years without an accident. That very day, Clark’s ship had an accident that killed him and everybody else aboard. Furthermore, while Burroughs was thinking about this crude example of the irony of the gods that evening, a bulletin on the radio announced the crash of an airliner in Florida, USA. The pilot was another captain Clark and the flight was Flight 23.

I looked at this and noticed that “William S. Burroughs II” (his full name) has—yes that is right—exactly 23 characters. What does this all mean?

## An Interesting Open Problem

We are after all about mathematics, so I decided to leave numerology behind and state one open problem that is not well known. It was thought of by Eduardo Casas-Alvero and is named for him. It is easy to state:

Let ${f}$ be a one-variable polynomial of degree ${d}$ defined over a field ${K}$ of characteristic zero. If ${f}$ has a factor in common with each of its derivatives ${f^{(i)}}$, ${i = 1,\dots,d-1}$, then ${f}$ must be a power of a linear factor.

See this for a nice survey of what is known, which includes a proof for ${d= 12}$. The lowest open ${d}$ currently is ${20}$, and the authors say attempting their calculation method for ${d=20}$ would be “utopic.” But solving it would prove the conjecture up through ${d = 23}$.

Let me note that “Casas-Alvero conjecture” has, yes—yes—twenty-three characters. By the way I selected this conjecture before I started to think about the 23 enigma—I did not search for a conjecture that fit.

## Open Problems

Is 23 the key to understanding the universe? Does it shed light on theory? The one that scares me a bit is:

$\displaystyle \text{Five hundred and twelve}$

has exactly 23 characters.

1. October 15, 2013 9:34 am

In the Realm of Riffs and Rotes

$23 = \text{p}_9^1 = \text{p}_{\text{p}_\text{p}^\text{p}}$

$512 = \text{p}_1^9 = \text{p}^{\text{p}_\text{p}^\text{p}}$

• October 15, 2013 9:38 am

There are pictures of 23 and 512 here.

2. October 15, 2013 10:25 am

23 is the smallest odd prime that is not a twin prime… (Wikipedia) :-)

• October 16, 2013 9:35 am

Of course the only reason you need the “odd prime” qualifier is because of the 2, which is right next to the 3.

3. October 15, 2013 12:04 pm

“Barbosa proves P != NP.” has exactly 23 characters.

The Proof is at: http://arxiv.org/ftp/arxiv/papers/0907/0907.3965.pdf.

4. October 15, 2013 12:58 pm

I have a prove that P=NP : “00100100110110111111000010100111001011011100001101100100110100110110110100010000111100111000010010011101011100010010111001011111111000101101111101100101001010011111110010011010000001100000000111110111010110100000111111111101011100011001110010001010111101000100010011100111000111000011111110101001011010110000010101011101110010001101111110101101000110110011100001011011010111001001110011000110110010110001001101000001011111011011101100110001010101011001000001011001011101000111101100011011100011000011011101010110”.

It has exactly 512 characters

• October 15, 2013 9:17 pm

Sorry! I do not understand your proof ;-)

• October 16, 2013 9:01 am

It requires the knowledge of commutative algebra, and some encryption theory ;). It is certain that there is a Turing, or Quantum or Sigma computer algorithm that can easily convert it into human readable form ;)

• October 16, 2013 9:38 am

Thanks! I think the first digits should be 10111, i.e. 23 ;-)

5. October 16, 2013 4:37 pm

Appreciation and thanks to Gödel’s Lost Letter and P=NP for 2^9-fold contributions to PROGRESS IN MATHEMATICS (23!).

—————
while true; do ( read -r; echo -n "\$REPLY" | wc -c ); done

• October 16, 2013 4:52 pm

A couple more 23-magical GLL topics: “PvsNP is a Hard Problem” and “Quantum Dynamical Paths” Best wishes for plenty more magical GLL essays!

6. October 18, 2013 7:16 am

As long as we’re thinking outside the box and off the wall, I might as well vent a bit of worry I have from time to time. Sometimes I wonder if we are always working from the right definition of a problem and the right idea of what counts as a solution. Do the notions of recognizing a set and generating a set, or computing a function and inverting a function, as important as they are, really give us the most natural, undistorted representation of the kinds of situations we pre-theoretically call problems in real life and scientific inquiry? Just something to think about …

October 18, 2013 9:10 am

“I might as well vent a bit of worry I have from time to time.’

Thanks for sharing your concern. For many of us it is less of a worry and more of a frustrating struggle to reconcile what we see with how we are instructed to see it!