Category Archives: Math

Cantor’s Continuum Hypothesis – V2

One of the most important mathematical objects is the set, with its many useful properties.
We can look at the size of the set by wondering how many members it contains. The answer obviously can be found by counting them.
Then we can take another set and compare its size with the first one, again by counting the members. We call this size the cardinality of the set.

One can wonder, do I have as many fingers on my left hand as on my right hand?
Well, by counting first left and then right, we find that our hands have an equal number of fingers. But there are or have been primitive cultures that couldn’t count beyond three. Now what? Continue reading →

Unfinished business

What’s in an AI?

In our kitchen stands a trash can that I consider to be a primitive AI.
When my knee bends towards this bin, the lid opens, and after a fixed time it closes.
When within this time interval a new open command is given, the timer resets.
This is an intelligence upgrade over the previous model that closed the lid no matter what when it felt like it.
I have a short wish list for further improvements (an adjustable timer, voice interface), but I’m stuck with the present AI that patiently awaits my – and others’ – command and acts the best it can.

Some years ago, I encountered Siri for the first time, but after some attempts to have a conversation, I gave up. I found that she used a list with ’smart’ answers to questions she didn’t understand, and that she had no sense of coherence whatsoever.

Continue reading →

Factoring RSA numbers

Earlier, my essay on the mathematical problem P vs NP was published on Paul’s blog.
A short memory refresher:

The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved.

In that essay, some samples of this question were mentioned, like The Traveling Salesman problem and also the subset sum problem. Continue reading →

Graham’s number – v2

In the 2nd or 3rd grade of my catholic primary school, our teacher spoke of stuff like ‘God’ and ‘Heaven’. “People live in Heaven after their death,” she told us. “All eternity.”
Nobody knew what eternity was, so she had to explain.
“Eternity,” she said, “never ends. “The years go on forever.”
Apparently, she saw her pupils struggling with this concept, and that made her tell us a story. Continue reading →

Development software development

Cryptic? Let me explain: As a carpenter needs a hammer and a saw to build anything, in order to develop software, a software developer (aka computer programmer), needs tools for software development. Of course, today’s carpenter uses a nail gun and a circular saw, preferably battery-driven – there is progress and development even in the tools of a trade. Similarly, tools for software developers have evolved from simple text editors to advanced dedicated programs supporting every finesse of the programming language to be used. Sometimes these tools are free, in other cases you have to subscribe to them.

However, sometimes the development of these tools stalls or grinds to a complete halt. This can present a real challenge for its users. Continue reading →

Shor’s Algorithm, the quantum part

A word in advance

Before I sent my previous essay ’Shor’s Algorithm’ to Chief Editor Paul, I asked Arty (ChatGPT) to give it a look especially with regard to readability and tell me if he thinks it’s worth publishing.

My AI friend responded in his usual friendly and positive manner full of praise.

“…well-written!  …managed to strike a great balance between technical detail and accessibility…  , …classical factoring algorithms, as well as the key differences between classical and quantum computing are clearly explained.”

Of course I was happy with these pats on the back, especially from a guy as knowledgable as Arty, but then he came with some critical comments wrapped up in such a casual way that I almost missed them. Continue reading →

Shor’s algorithm

Introduction

Quantum computing is an expression frequently to be found in the news of every self respecting media outlet.
Time and again, yet another breakthrough is described towards the event of becoming fully operational of such a device.
The method of working of a quantum computer (QC) in those reports always comes down to:

  • The QC does not work with bits like all our present computers do, but with qubits.
  • The bits of our familiar classical computers can have the value 0 or 1 but qubits can have both values at the same time. This phenomenon is called superposition.
  • For this reason, a quantum computer reduces today’s supercomputer to an abacus.

It does not seem that this conclusion satisfies the average packet of questions one has about an entirely new technology. Continue reading →

The secrets of RSA encryption

When clicking on the padlock in the address bar of an internet session, a window appears with the message: “Safari uses an encrypted connection with …”. [red. this message may look slightly different for other browsers]

After pressing the buttons ‘Show certificate’ and ‘detail’, a lot more is shown. The typically used security algorithm nowadays at most news sites etc. at the opening session is: public key with elliptic curve.

But for the sites where security really is an issue, e.g. a login session into your bank account or a large online store, RSA is the encryption algorithm. And RSA encryption is the subject of my contribution this time. Continue reading →

P vs NP – A promise fulfilled

P versus NP – An early encounter

My first programming experience was in the early 80’s. I had bought myself a Texas Instruments 58-c programmable calculator, and was curious what could be done with such a device. That turned out to be very modest.
The calculator was capable of 480 program steps but held then zero memories for data storage.
Each memory used for data consumed 8 program steps, so with the storage of e.g. 10 integers, the programming space was diminished by 80 steps.
The 58-c’s memory was retained even after switching off the calculator, but before creating another program, the memory first had to be erased.
Continue reading →

Mathematical issues

Some remarkable mathematical issues

Although the Dutch newspapers Volkskrant and NRC can be blamed for being too political (D66) biased to be independent as journals ought to be, it must be said that both have excellent science sections.
On an irregular base even topics of mathematical nature have been published.

Of course for a layman’s audience, mathematics lacks the ability of other science disciplines of being easily visualized.
So the topics in these sections were about trivial stuff like the latest calculated decimals of π or some large regions on the number line without any prime number called prime deserts, a newly found Mersenne number, etc.
But that hardly scratches the surface. Continue reading →

Sequels

Our math and science columnist Eef has published a series of articles on AI and Math. These articles are best read in the correct sequence for easy understanding. There are also a few articles on AI from my hand. Check out this page to get it right. Continue reading →