Covering set

In mathematics, a covering set for a sequence of integers refers to a set of prime numbers such that every term in the sequence is divisible by at least one member of the set. The term "covering set" is used only in conjunction with sequences possessing exponential growth.

Sierpinski and Riesel numbers

The use of the term "covering set" is related to Sierpinski and Riesel numbers. These are odd numbers|odd] natural numbers for which the formula or produces no prime numbers. Since 1960 it has been known that there exists an infinite number of both Sierpinski and Riesel numbers (as solutions to families of congruences based upon the set but, because there are an infinitude of numbers of the form or for any, one can only prove to be a Sierpinski or Riesel number through showing that every term in the sequence or is divisible by one of the prime numbers of a covering set.
These covering sets form from prime numbers that in base 2 have short periods. To achieve a complete covering set, Wacław Sierpiński showed that a sequence can repeat no more frequently than every 24 numbers. A repeat every 24 numbers give the covering set, while a repeat every 36 terms can give several covering sets: ; ; and.
Riesel numbers have the same covering sets as Sierpinski numbers.

Other covering sets

Covering sets also exists for bases other than 2.

b	smallest k such that gcd = 1 and k×''bn''+1 has covering set	covering set for k×''bn''+1	smallest k such that gcd = 1 and k×''bn''−1 has covering set	covering set for k×''bn''−1
2	78557		509203
3	125050976086		63064644938
4	66741		39939
5	159986		346802
6	174308		84687
7	1112646039348		408034255082
8	47		14
9	2344		74
10	9175		10176
11	1490		862
12	521		376
13	132		302
14	4		4
15	91218919470156		36370321851498
16	66741		33965
17	278		86
18	398		246
19	765174		1119866
20	8		8

Covering sets are also used to prove the existence of composite generalized Fibonacci sequences with first two terms coprime, such as the sequence starting with 20615674205555510 and 3794765361567513.
The concept of a covering set can easily be generalised to other sequences which turn out to be much simpler.
In the following examples + is used as it is in regular expressions to mean 1 or more. For example, 91⁺3 means the set.
An example are the following eight sequences:

/ 9 or 32⁺01
/ 9 or 41⁺51
/ 9 or 51⁺81
/ 9 or 65⁺23
/ 9 or 91⁺3
/ 9 or 94⁺9
/ 9 or 95⁺9
/ 9 or 98⁺23

In each case, every term is divisible by one of the primes from the set. These primes can be said to form a covering set exactly analogous to Sierpinski and Riesel numbers. The covering set is found for several similar sequences, including:

/ 9 or 42⁺07
/ 9 or 4⁺07
/ 9 or 81⁺51
9175·10ⁿ + 1 or 91750⁺1
10176·10ⁿ − 1 or 101759⁺
/9 or 371⁺
/3 or 40703⁺
/9 or 8917⁺

Also for bases other than 10:

521·12ⁿ + 1 or 3750⁺1 in duodecimal
/11 or 991⁺ in duodecimal
/11 or 2X27⁺ in duodecimal
376·12ⁿ − 1 or 273E⁺ in duodecimal

The covering set of them is
An even simpler case can be found in the sequence:

/ 99 or ⁺7

Here, it can be shown that if:

w is of form : ⁺7 is divisible by 7
w is of form : ⁺7 is divisible by 13
w is of form : ⁺7 is divisible by 3

Thus we have a covering set with only three primes. This is only possible because the sequence gives integer terms only for odd n.
A covering set also occurs in the sequence:

/ 9 or 381⁺.

Here, it can be shown that:

If, then is divisible by 3.
If, then is divisible by 37.
If, then algebraically factors as.

Since can be written as 23⁺, for the sequence 381⁺, we have a covering set of – a covering set with infinitely many terms.
The status for /9 is like that for 3511808×63ⁿ + 1:

If, then is divisible by 37.
If, then is divisible by 109.
If, then algebraically factors as

Thus we have a covering of or – a covering set with infinitely many terms.
A more simple example is 4×9ⁿ − 1, it is equal to ×, thus its covering sets are and, more generally, if k and b are both r-th powers for an odd r > 1, then k×''bn'' + 1 cannot be prime, and if k and b are both r-th powers for an r > 1 then k×''bn'' − 1 cannot be prime.
Another example is 1369×30ⁿ − 1, its covering is