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 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 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.
bsmallest k such that gcd = 1 and k×bn+1 has covering setcovering set for k×bn+1smallest k such that gcd = 1 and k×bn−1 has covering setcovering set for k×bn−1
278557509203
312505097608663064644938
46674139939
5159986346802
617430884687
71112646039348408034255082
84714
9234474
10917510176
111490862
12521376

Covering sets are also used to prove the existence of composite generalized Fibonacci sequences with first two terms coprime.
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:
In each case, every term is divisible by one of the primes. 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:
Also for bases other than 10:
The covering set of them is
An even simpler case can be found in the sequence:
Here, it can be shown that if:
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:
Here, it can be shown that:
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 3511808×63^n+1:
Thus we have a covering of or – a covering set with infinitely many terms.
A more simple case is 4×9^n−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×b^n+1 cannot be prime, and if k and b are both r-th powers for an r>1 then k×b^n−1 cannot be prime.
Another case is 1369×30^n−1, its covering is