Puzzle 1280 Plenty of Sierpinski and Riesel numbers, and the conversion between them. A second Web Calculator.

Dedicated to Wilfrid Keller

On August 18, 2026, Carlos Rivera wrote:

This puzzle is my way to introduce a second Web Calculator. This time the Calculator is devoted to generate other Sierpinski or Riesel numbers starting with a first one. The Calculator also shows how to calculate a Riesel number from a Sierpinski one and viceversa.

I have chosen the name of this Calculator as "Calculador3NCS". The "3" indicates that this Calculador executes 3 different processes. The "NCS" indicates that this Calculador needs as input only two kind of data: a) The N value (S or R) and b) the CS or Covering set of the N value.

And that's all the Calculator needs to be inputted. You need not to declare if the N value inputted is Sierpinski or Riesel.

After that, you need only press the "Calcular" button and the results will be dispalyed in three output windows, in the following  order:

As a preliminary step the Calculador3NCS compute & shows the M & P values corresponding to the CS inputted. This step is necessary because both values, M & P, will be used in the following steps.

1) In the upper window, will be displayed only 10 of an infinite N numbers that share the same CS as the first & inputted one. The formula used here is: Nj = N + j·P, j=2, 4, 6, ... ∞. (like a stairway to heaven...)

2) In the second window will be displayed the M distinct numbers N of the Cycle of Keller starting in the first one inputted. Again all of these M numbers N share the same CS. In this windows the minimum and the maximum values (Nmin & Nmax, respectively) will be shown with colour green & red, respectively. The formula used here is a recursive and modular one:
N → (2·N + P) mod (2·P).

All of these M values are distinct N and there are no more because once the list is completed, the values are repeated in the same order (like the links of a bicycle chain...)  

3) In the third window will be displayed the Riesel number if your N inputed was a Sierpinsiki one, and viceversa. Again both of these numbers share the same CS. The formula used in this Conversion is R = 2·P − S (or S = 2·P − R).

This Calculador3NCS can be found here: https://www.primepuzzles.net/Calculador3NCS.html and also in the Links page, in the section of "Mathematical Web Tools", in the 11th place.

And now two questions:

Q1. May it happen that the "stairway to heaven" share at least one common element with the "links of a bicycle chain", other than the initial one if this is set as Nmin of a Keller's Cycle, for both sequences?

Q2.In any Keller Cycle how big coud it be (Nmax-Nmin)?

Addendum:

I take the opportunity to announce that my first Web CS-CalculadorV5, [https://www.primepuzzles.net/CS-CalculadorV5.html] has been improved and now is in operation the new one https://www.primepuzzles.net/CS-CalculadorV6.html, while the V5 has been discontinued.

Why?...The previous one was not preprared to handle Brier numbers; while the new one can handle the most of the known ones, as compiled by W. Keller. In defining which restrictions of my CS-CalculadorV5 should be changed, was very helpful the Catalog of Brier numbers recently uptaded by Wilfrid Keller.

At the end I only had to change only two restrictions:
1) To change the two fixed primes of the CSs from {3, 5} to only one fixed prime {3}.
2) To incresae the sizes range of the CSs from 6-11 to 6-12.

After these two little changes my new CS-CalculadorV6 was able to compute the CS of 487 (86.34%) of the 564 CS of the 282 Brier numbers listed in the Keller's Catalog mentioned. The Brier numbers out of my CS-CalculadorV6, are out by one of the two following reasons: These Briers numbers a) use CS Sizes greater than 12 or b) use CSs with Primes greater than 1321.

From the old CS-CalculadorV5, remains the 24 primes base-list used in the Combinatory work. Also, I did'nt let the sizes to be extended to the wider range 6-13 just to keep the new Calculador as a light one (short time of response).

If in the future, if I'm able to optimize the algoritm behind scene I could try to include the size 13 and perhaps to add one more prime to the base-list of them. But let me tell you that only 4 of the 282 Brier numbers in the Keller's catalog use size 13 while 60 CSs use the prime 61681... . In any case this possibility belongs to the future.

A succesful Brier example:

This will be a process of three steps.

Step 1:
Inputting the smallest known Brier number (Clavier, 2013), 3316923598096294713661 (22 digits), into the CS-CalculadorV6.
a) after pressing the Sierpinski button, and the Calcular button you get: {3,5,13,17,97,241,673}, size=7, M=48, P=52153970115

Step 2:
b) after pressing the Riesel button, and the Calcular button you get: {3,7,11,19,31,37,41,73,109,151,331,1321}, size=12, M=180, P= 108435121946131097271.

Step 3:
If you want to know the global M & P of this Brier number you have to use the Calculador3NCS, inputing 3316923598096294713661 & the combined CS from the two CSs computed before,  CS= {3,5,13,17,97,241,673,7,11,19,31,37,41,73,109,151,331,1321}. After pressing the Calcular button you get:
M = 720, P = 1885107369798300628981297352055.
 
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On Aug 27, 2026, Carlos Rivera announced that his new version of the CS Calculador is now availabe as CS-CalculadorV7.

It was expanded in three directions in order to cope with the most of the published Brier integers:

a) It was added the prime 61681 to the original list of primes-base of 24 primes
b) It was added the size 13 for the CS's seek in the combinatorial work
c) The limit for the M allowed was changed from 200 to 400.

The explanation section was expanded to guide how to deal with Brier numbers.

Moreover, the second Online Calulator 3NCS has been upgraded to 3NCS2, totally in English and with an "Explanation" section whose core is the guide to deal with Brier numbers. 


Enjoy them

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On Aug 30, 2026, C. Rivera announced his new version of the CS Calculador is now availabe as CS-CalculadorV8. It mainly has improvements in the way of handle & results for calculating the CS of the Brier numbers. Please read carefully the Explanaion section.

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