Puzzle 1279 A Covering Sets Calculator

Dedicated to Arkadiusz Wesolowski & Emilia Gurasatti.

On August 5, 2026, Carlos Rivera wrote:

This puzzle is my way to introduce my own Calculator of Covering Sets, for a given integer N, type S or R, when the inputted N & type has CSs.

Let me share a little story — necessary to understand how this was done:

At the end of May, 2026, I began with the idea of building a tool to verify the CS published in my site for Sierpinski or Riesel integers by several authors. This idea was strongly inspired by the Multi-CS (3 & 4) integers published by Emilia Gurasatti in my Problem 92.

But right away I faced the first obstacle: there was no public calculator online for doing this!

Friends told me the same thing: the problem is computationally very hard or complex because of the huge prime base to search in, and the enormous number of combinations to test.

In short, such a code seemed computationally unfeasible.

Perhaps because fortunately I’m not a Mathematician but an Engineer, I thought: what if I could reduce the complexity and still produce an “acceptable” Calculator? As Mexican Engineers we often say: “If there is no bread, then tortillas.”

While reading everything I could about lists of N (S or R) values and their CSs, I noticed three key facts:
a) Most CS were made of 6 to 11 primes
b) The list of all distinct primes involved in these CS was only 24.
c) Almost all of the CS published starts with 3 & 5 in the first and second place, respectively.

That’s when I said: “I have found the tortillas for this problem.”

So I switched direction: what if I build a code for this reduced task? I has very clear in my mind the main mathematical steps for doing this task.

Then came the second obstacle: I don't know how to program in any modern language. My last programs were written in the beautiful (for me) but old and discontinued Ubasic by Yuji Kida. Since Windows 64 made it difficult to use, I stopped programming in it, around some 10 years ago.

I tried a third path: learning Python online. But I soon abandoned the idea — it wasn’t going to be easy for me.

Then I turned to a fourth, more obscure path: what if AI support could generate the Python code for me? A program that takes N and its Type (S or R) and outputs its CS, if any.

After wrestling with the idea in my mind, I thought: I lose nothing by asking. So I chose ChatGPT and posed the question: “Can you make a Python code for…?” The first answer was yes, but with a warning: the response time could be very, very long.

I replied: what if I give you three ideas to cut down this huge calculation time? ChatGPT said: “We can try, under your responsibility.”

And so we began. By early June I switched to Copilot, for reasons not important to mention here. By early July, the work was finished in the Python part. The rest of the work was to translate the Python code to JS for having a Web page. This part was started with Copilot and finished completely with Claude.

The CS-CalculadorV5.html

The code works with two inputs: N and Type (S or R) and two buttons: Calcular and Borrar. The list of 24 primes I defined, is embedded in the code.

There are two early sieves for the inputted N value and its type: a) N is refused if it is even; b) N is refused in his type, is there is a n<1000 such that N*2^n+/-1 is prime.

After that, the CS-CalculatorV5, generates all the combinations of the 24 prime list, in packets from from 6 to 11 primes, using the rule: the primes 3 & 5 go first. All the CS produced are tested as "covering" and as "minimals" and another restriction: the combination of primes need to have a Modulus M<= 200. If the combination pass these three tests, the combination goes to the output, as a CS together with its Module M & the Producf of its primes P.

End of the story.

You can find the Calculator in the following link, https://www.primepuzzles.net/CS-CalculadorV5.html  and also in the Links page, in the section of "Mathematical Web Tools" listed there in the tenth place.

Whenever you go to the Calculator page, you will find in the bottom, the basic instructions and details of the calculator's origin.
Now, the questions.

Q1. What is the first N (S or R) that has a CS with only 5 distinct primes, if there is one?

Q2. What is the first N (S or R) that has a CS with only 10 distinct primes?

Q3.  What are the minmal S or R integers that need 12 or more distinct primes in its CS?

Q4. What is the minimal S or R integer having five distinct CS?

Q5. The Multi-CS ntegers are important? Why? Or they are another wrong twist of the screw?

Q6. Regarding the 24 prime list: {3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 181, 241, 257, 331, 433, 557, 631, 673, 1321}, do you devise any other set more convenient for this purpose?