Williams Numbers

One of the properties of the number 26342 (my diurnal age as of Monday, May 17th 2021) is that it is a member of OEIS A204322

A204322

Numbers n such that \( 4 \times 5^n + 1\) is prime.

At first glance, this didn't look all that interesting but further investigation led me to discover that \( 4 \times 5^{26342} + 1\) produces what is called a Williams prime of the second kind. These are primes of the form:$$(b-1)\cdot b^{n}+1 \text{ for integers } b \geq 2 \text{ and } n \geq 1$$In this particular case, \(b=5\) and thus \( 4 \times 5^n + 1\). The sequence begins:

0, 2, 6, 18, 50, 290, 2582, 20462, 23870, 26342, 31938, 38122, 65034, 70130, 245538 

These primes becomes very large indeed with increasing \(n\). Let's consider a small value of \(n\) such as \(n=2\). Here we have \(5 \times 6^2+1 = 181\) which is prime. Numbers of the form \((b-1)\cdot b^{n}+1 \) that are not prime are simply called Williams numbers of the second kind. 

Wikipedia has a list of the bases from 2 to 30 along with the indices that produce primes. The Williams primes of the second kind base 2 are exactly the Fermat primes. As of September 2018, the largest known Williams prime of the second kind base 3 is: $$2×3^{1175232}+1$$So what are Williams numbers and primes of the first kind? A Williams number base \(b\) is a natural number of the form: $$(b-1)\cdot b^{n}-1 \text{ for integers } b \geq 2 \text{ and } n \geq 1$$The Williams numbers base 2 are exactly the Mersenne numbers. A Williams prime is a Williams number that is prime. For base 5, the initial Williams primes of the first kind are:

1, 3, 9, 13, 15, 25, 39, 69, 165, 171, 209, 339, 2033, 6583, 15393, 282989, 498483, 504221, 754611, 864751, ...

Looking at the case of \(n=3\), we see that \(4 \times 5^3-1 = 499\) which is prime. 

A Williams number of the third kind base \(b\) is a natural number of the form:$$(b+1)\cdot b^{n}-1 \text{ for integers } b \geq 2 \text{ and } n \geq 1$$The Williams numbers of the third kind base 2 are exactly the Thabit numbers. A Williams prime of the third kind is a Williams number of the third kind that is prime.

A Williams number of the fourth kind base \(b\) is a natural number of the form:$$(b+1)\cdot b^{n}+1 \text{ for integers } b \geq 2 \text{ and } n \geq 1$$A Williams prime of the fourth kind is a Williams number of the fourth kind that is prime, such primes do not exist for \(b\equiv 1 \bmod {3}\).

So to summarise, a Williams number of whatever kind will conform to this pattern:$$(b \pm 1)\cdot b^{n} \pm 1 \text{ for integers } b \geq 2 \text{ and } n \geq 1$$