Covering Set - Other Covering Sets

Other Covering Sets

Covering sets are also used to prove the existence of composite Fibonacci sequences (primefree sequence).

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 {913, 9113, 91113, 911113…}

An example are the following three sequences:

• (82·10n + 17) / 9 or 91+3
• (85·10n + 41) / 9 or 94+9
• (86·10n + 31) / 9 or 95+9

In each case, every term is divisible by one of the primes {3, 7, 11, 13}. These primes can be said to form a covering set exactly analogous to Sierpinski and Riesel numbers.

An even simpler case can be found in the sequence:

• (76·10n − 67) / 99 (n must be odd) or (76+7

Here, it can be shown that if:

• w is of form 3 k (n = 6 k + 1): (76)+7 is divisible by 7
• w is of form 3 k + 1 (n = 6 k + 3): (76)+7 is divisible by 13
• w is of form 3 k + 2 (n = 6 k + 5): (76)+7 is divisible by 3

Thus we have a covering set with only three primes {3, 7, 13}. This is only possible because the sequence gives integer terms only for odd n.

A covering set also occurs in the sequence:

• (343·10n − 1) / 9 or 381+.

Here, it can be shown that:

• If n = 3 k + 1, then (343·10n − 1) / 9 is divisible by 3.
• If n = 3 k + 2, then (343·10n − 1) / 9 is divisible by 37.
• If n = 3 k, then (343·10n − 1) / 9 is algebraic factored as ((7·10k − 1) / 3)·((49·102k + 7·10k + 1) / 3).

Since (7·10k − 1) / 3 can be written as 23+, for the sequence 381+, we have a covering set of {3, 37, 23+} – a covering set with infinitely many terms.