Calculador de covering sets
Sierpinski / Riesel / Brier

Explanation of the CS-Calculador.html

Last news: 9/21/2026
This will most likely be the last version of the CS-Calculator. This is because, after the latest changes to V8, we are now able to calculate the CS for ALL the Brier integers listed in W. Keller's Catalog.
For the user, nothing changes, but the JS code was modified to handle all 283 Briers in the Catalog.
For that same reason, the URL for this CS-Calculator will simply be https://www.primepuzzles.net/CS-Calculador.html, and it will no longer change even if small modifications are made in the future, for the users' convenience.
Special thanks to Claude, who helped develop this final stable version. This AI found a strategy that keeps the processing time for almost all Briers under 2 minutes, with only one exception, where it can reach up to 2.5 minutes.

This page computes minimal covering sets (CS) for an input number (k), Sierpinski (S) or Riesel (R) or Brier (B), subject to the restrictions mentioned below.

Usage

1. Enter number k.
2. Select its type (S, R or B).
3. Press Calcular.

The output (Results) includes

1. The input number and type.
2. The quantity of CSs found (if any).
3. All the CS found:
   • a) The list of primes composing each CS
   • b) Its Size (quantity of primes in each CS)
   • c) M: Its Module = Least common multiple of the multiplicative modular orders of two for those primes.
   • d) P: The product of all the primes in each CS.
4. The list of sizes of all the CS found.
5. The execution time.

If B (Brier) is selected, the page computes k automatically first as Sierpinski and then as Riesel in a single run, printing the CS/size/M/P results for each. If a CS is not found for either case, the calculation stops with a "CS not found for k as Sierpinski/Riesel" message. If CS are found for both cases, the page also reports their union — called CSglobal — with its own Size, M and P. Since CSglobal is a union of two independent CS, the M<=400 restriction below does not apply to it; its M can reach up to 720.

Here is one output example over a Brier number by Yves Gallot and Carlos Rivera:

 Número: 721413784980345829736061421921 (B, 30 dígitos)
--- Como Sierpinski (S) ---
Número: 721413784980345829736061421921 (S, 30 dígitos)
Covering sets encontrados: 1
Detalle: [{3,5,11,17,31,41,61,331,1321,61681}, 10, 120, 5865303216826207605]
Sizes: 10
Tiempo: 4.184 s
--- Como Riesel (R) ---
Número: 721413784980345829736061421921 (R, 30 dígitos)
Covering sets encontrados: 2
Detalle: [{3,7,13,19,37,73,97,109,241,257}, 10, 144, 9174644185821387]; [{3,7,13,19,37,73,97,241,257,433,673}, 11, 144, 24528200803174537287]
Sizes: 10, 11
Tiempo: 6.885 s
--- CSglobal ---
S1+R1: CSglobal: {3,5,7,11,13,17,19,31,37,41,61,73,97,109,241,257,331,1321,61681}, Size: 19, M: 720, P: 17937356685444681190725085037016045 (Rivera)
S1+R2: CSglobal: {3,5,7,11,13,17,19,31,37,41,61,73,97,241,257,331,433,673,1321,61681}, Size: 20, M: 720, P: 47955111691272927533100975280291822545 (Gallot)
Tiempo CSglobal: 0.000 s
Tiempo total: 11.069 s

If you wish you can accumulate several independent results in the same box. But if you want to clean all the fields you can use the "Borrar" button.

If k is such that k*2^n+/-1 is prime for some n<1000 you get a message in such a sense, which means that k is not Sierpinski or Riesel. This is just a small early sieve to discard false integers S or R. But if the output is "Not CS found" it might be caused by the limitation of the algorithm.

If k is not odd you get a message in such a sense.

Code Restrictions

1. k must be odd.
2. Types limited to Sierpinski (S) or Riesel (R) or Brier (B) numbers. If you introduce a Brier number (B), the page computes its two covering sets in the same run: first as Sierpinski, then as Riesel, plus their CSglobal union if both are found.
3. The search for CS is restricted to combinations of the following twenty-five primes: {3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 181, 241, 257, 331, 433, 577, 631, 673, 1321, 61681}. If no CS is found this way, the search is automatically retried with eight additional primes — {193, 641, 1153, 6337, 38737, 65537, 6700417, 487824887233} — needed to cover a handful of special cases in Keller's Catalog.
4. CS sizes computed are limited to the Range: 6–13, being the first prime in them always and only the prime 3.
5. M<=400, raised to M<=720 for the second search pass described in point 3 above. This limit does not apply to the CSglobal reported for Brier numbers.
6. Moreover, if the S and or R happens to be Multi-CS, this code will produce all the CSglobal as combinations of CS for S and R exist.

These restrictions imposed to my CS-Calculator derive from my personal observations of:

1. The 50 CSs reported at irvinemclean.com/maths/siernums.htm
2. Analysis by my Python code, of 15,000 S and 15,000 R numbers in OEIS:
   • i. oeis.org/A076336/b076336.txt (Sierpinski, T. D. Noe & A. Wesolowski).
   • ii. oeis.org/A101036/b101036.txt (Riesel, P. CAMI & A. Wesolowski).
3. The Multi-CS S & R numbers published by Emilia Gurisatti in:
   • iii. github.com/emiguri/A394518
   • iv. github.com/emiguri/A394519
4. The Multi-CS S & R numbers provided by A. Wesolowski in: Prob 92 AW.txt
5. The Catalogue of Brier integers, by W. Keller, prothsearch.com/Catalogue.txt

Regarding the limit for M as <=400, I can say that none of the 15,000 Sierpinski or the 15,000 Riesel reported in OEIS, exceeded that M limit imposed to the search for CS in my Code. The same I can say about the list of 144 Sierpinski reported by I. McLean in the link to his work above.

But... as with any publicly available tool, occasional errors may arise; notifications via email regarding any issues will be sincerely appreciated.

Use it with confidence and thank you for sharing in this mathematical exploration.

Notes

1. A CS is "minimal" if all primes listed are necessary for the list to remain a Covering Set.
2. The overall twenty-five-primes list used in the Combinatorial work coincides with all revised sources but is not guaranteed to cover all possible S or R or Brier numbers you may input. The aim is a light calculator, producing exhaustive results in no more than 2.5 minutes.
3. It is recommended not to switch tabs while a calculation is running, since some browsers may pause it in the background.

Acknowledgments

Special thanks to ChatGPT, Copilot & Claude for assistance in building the Python script and converting it to JavaScript for this page.

Final responsibility for the project rests with the author.