Calculador de covering sets
Sierpinski / Riesel / Brier
Enter an odd integer k and select the type. First, k is examined by testing the primality of k·2ⁿ ± 1 for n = 1 to 1000; if it survives, its minimal covering sets are sought.
Explanation of the CS-CalculadorV8.html
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
- Enter number k.
- Select its type (S, R or B).
- Press Calcular.
The output (Results) includes
- The input number and type.
- The quantity of CSs found (if any).
- 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.
- The list of sizes of all the CS found.
- 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 Riesel, original by Emilia Gurisatti:
Número: 16107442173648029 (R)
Covering sets found: 4
Details: [{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
Time: 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 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 your get a message in such a sense.
Code Restrictions
- k must be odd.
- 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.
- 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}.
- CS sizes computed are limited to the Range: 6–13, being the first prime in them always and only the prime 3.
- M<=400 (this limit does not apply to the CSglobal reported for Brier numbers).
- 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. Please run the B = 721413784980345829736061421921 to see this in detail
These restrictions imposed to my CS-Calculator derive from my personal observations of:
- The 50 CSs reported at irvinemclean.com/maths/siernums.htm
- 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).
- The Multi-CS S & R numbers published by Emilia Gurisatti in:
- The Multi-CS S & R numbers provided by A. Wesolowski in: Prob 92 AW.txt
- 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
- A CS is "minimal" if all primes listed are necessary for the list to remain a Covering Set.
- 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 two minutes.
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.