Problems & Puzzles: Puzzles

Problem 92. Multiple covering sets for Sierpinski and for Riesel integers.

On April 2, 2025, Emilia Gurisatti sent the following interesting problem.

She is "estudiante de segundo año de la Licenciatura en Ciencias Matemáticas en la Universidad de Buenos Aires, Argentina"

Emilia found several (27) Sierpinski integers with 3 distinct covering sets. Emilia got the same kind of results for other several (23) Riesel integers.

Her results are published, since Mar 23, 2026, in:

https://oeis.org/A394518 & https://github.com/emiguri/A394518/blob/main/b394518.txt

and

https://oeis.org/A394519 & https://github.com/emiguri/A394519/blob/main/b394519.txt

On top of that she got two more results not already published but sent to me by mail on April 2, 2026.

Sierpinski integer with 4 covering sets, distinct:

 218164505156603
Covering sets: {3,5,7,13,19,37,73}, {3,5,7,13,17,241}, {3,5,7,13,19,37,109} y {3,5,7,13,97,241,257}

Riesel integer with 4 covering sets, distinct:

 16107442173648029
 Covering sets  {3,5,7,13,19,37,73}, {3,5,7,13,17,241}, {3,5,7,13,19,37,109} y {3,5,7,13,97,241,257}

Wilfred Keller, who was also contacted and consulted during the development of Emilia's work, sent the following comment:

"The study of multiple covering sets presented by Emilia Gurisatti took me  by surprise. Long ago, I had encountered two "double" coverings, but had considered them as an annoying anomaly. They are contained in an unpublished draft of 1993. I didn't expect a systematic investigation of them..."

Emilia also make me notice that Arkadiusz Wesolowski, in 2015-2018 produced two covering sets for each one of 24 Serpinski integers (A263391) and also two covering sets for each one of 24 Riesel integers (A263392).

Here are the Covering sets found by Wesoloski for these 48 integers, and sent by him specially for this publication.

 Emilia question is:

Q1. Can you find Sierpinski or Riesel integers with more than 4 covering sets?

My (CR) questions are the following ones:

Q2. Can you find Brier integers with 2 or more covering sets?

Q3. Can you give your opinion about the importance/significance of investigating and producing múltiple covering sets for individual S/R/B numbers?
 

From April 18 to 25, 2026, contributions came from Arkadiusz Wesolowski, Emilia Gurasatti

***

Arkadiuz wrote:

Answer to Q3.
Let k be a Sierpiński number that has two covering sets, namely A and B, that are disjoint, and let U be the union of these two sets. If k > (p*q^2 - 1)/2, where p is the second largest prime in U and q is the largest prime in U, then k*2^n + 1 has at least three distinct prime factors for all positive integers n.
Example:
k = 132430318140059718398963877387394625974301919973914\
2231792024697220514811752089042898559422886932054246281\
43798648568555082755511814932143700294388691289 (153 digits)
Covering sets:
A = {5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 109, 151, 241, 331},
B = {3, 23, 29, 43, 67, 71, 89, 97, 113, 127, 257, 281, 337, 353, 397, 433, 449, 577, 641, 673, 683, 1153, 1429, 2113, 2689, 5153, 5419, 6337, 7393, 20857, 38737, 65537, 86171, 122921, 312709, 599479, 6700417, 7416361, 15790321}.
k*2^1 + 1 has both 3 and 31 as prime factors.
k*2^2 + 1 has both 7 and 127 as prime factors.
k*2^3 + 1 has both 3 and 61 as prime factors.
k*2^4 + 1 has both 5 and 29 as prime factors.
k*2^5 + 1 has both 3 and 7 as prime factors.
...
Analogously, let k be a Riesel number that has two covering sets, namely A and B, that are disjoint, and let U be the union of these two sets. If k > (p*q^2 + 1)/2, where p is the second largest prime in U and q is the largest prime in U, then k*2^n - 1 has at least three distinct prime factors for all positive integers n.
Example: 
k = 216138667663290358832800493328913904693367459100439\
7667247382156052061926161528977758304879313344338677262\
806049227643609138472430270365245414447323640007 (154 digits)
Covering sets are the same as above.
k*2^1 - 1 has both 3 and 7 as prime factors.
k*2^2 - 1 has both 31 and 127 as prime factors.
k*2^3 - 1 has both 3 and 5 as prime factors.
k*2^4 - 1 has both 7 and 29 as prime factors.
k*2^5 - 1 has both 3 and 17 as prime factors.
...
It is easy to show that there exist infinitely many Sierpiński and Riesel numbers with this property.

This statement is trivial.
We assume that k > (p*q^2 - 1)/2 to exclude the cases where, for some odd k, k*2 + 1 has two distinct prime factors and at least one of those factors is a prime power.

It can be show by constructing a fixed arithmetic progression modulo 2*d, where d is the product of the primes in U.

***

Emilia wrote:

I have found a Brier number that has at least two covering sets.

 k = 773553855548262449239841 (24 digits)

Covering set 1:
 C1 = [3, 5, 17, 97, 241, 257, 673]
P1 = 1031043870735
e_i1 = [2, 4, 8, 48, 24, 16, 48]
e1 = 48

C2 = [3, 7, 11, 13, 19, 31, 37, 41, 61, 73, 109, 151]
P2 = 196658330577361653
e_i2 = [2, 3, 10, 12, 18, 5, 36, 20, 60, 9, 36, 15]
e2 = 180

C = [3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 151, 241, 257, 673]
P = 67587788790255388691025974985
e_i = [2, 4, 3, 10, 12, 8, 18, 5, 36, 20, 60, 9, 48, 36, 15, 24, 16, 48]
e = 720

 Covering set 2:
 C1 = [3, 5, 17, 97, 241, 257, 673]
P1 = 1031043870735
e_i1 = [2, 4, 8, 48, 24, 16, 48]
e1 = 48

C2 = [3, 7, 11, 13, 19, 31, 37, 41, 61, 73, 109, 331]
P2 = 431085479609978193
e_i2 = [2, 3, 10, 12, 18, 5, 36, 20, 60, 9, 36, 30]
e2 = 180
C = [3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 61, 73, 97, 109, 241, 257, 331, 673]
P = 148156013838241944746553627285
e_i = [2, 4, 3, 10, 12, 8, 18, 5, 36, 20, 60, 9, 48, 36, 24, 16, 30, 48]
e = 720

The first covering set was originally used by Arkadiusz to cover k = 56284389701328043058161 (Wesołowski 5). The second one was originally used by Vantieghem to cover k=11615103277955704975673 (Vantieghem 2).

Would you like to verify it?

***

 

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