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 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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