Calculador de covering sets
Sierpinski / Riesel

Explanation of the CS-Calculator

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

Usage

1. Enter number N.
2. Select its type (S or R).
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.

Here is one output example, over a Riesel, original by Emilia Gurisatti:

Número: 16107442173648029 (R)
Covering sets encontrados: 4
Detalle: [{3,5,7,13,17,241}, 6, 24, 5592405]; [{3,5,7,13,19,37,73}, 7, 36, 70050435]; [{3,5,7,13,97,241,257}, 7, 48, 8200768485]; [{3,5,7,13,19,37,109}, 7, 36, 104595855]
Sizes: 6, 7, 7, 7
Tiempo: 1.507 s

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

If N is such that N*2^n+/-1 is prime for some n<1000 you get a message in such a sense, which means that N 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", the most probable is that the inputted N is not a S or R too.

If N is not odd your get a message in such a sense.

Code Restrictions

1. N must be odd.
2. Types limited to S or R.
3. Search for CS is restricted to combinations of the following twenty-four 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}.
4. CS sizes computed are limited to the Range: 6–11, being the first two of them the primes 3 and 5.
5. M<=200.

These restrictions imposed to my CS-Calculator derive from my 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

Regarding the Range 6-11 for the sizes of the CS I must say that only 3 Sierpinski integers of the first 15,000 reported in OEIS pages, needed the superior limit (11); while none of the 15,000 Riesel reported in OEIS pages needed that superior limit, according to the results of my Python code used to analyze them.

Regarding the limit for M as <=200, I can say that none of the 15,000 Sierpinski or the 15,000 Riesel reported in OEIS, exceeded the M limit imposed to the search for CS in my Code. The same I can say about the list of 144 Sierpinskis 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. Although Sierpinski numbers with CS size ten have not been observed, this size was included in the computations, just in case.
3. The overall twenty-four-primes list used in the Combinatorial work coincides with all revised sources but is not guaranteed to cover all possible S or R numbers you may input. The aim is a light calculator, producing exhaustive results in no more than two seconds.

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.

CS-CalculadorV5.html, Edition: August 2026
Carlos Rivera, assisted by ChatGPT, Copilot & Claude
For The Prime Puzzles & Problems Connection.
cbrfgm@gmail.com