Honaker: Primes and Problem

Figure 1

G. L. Honaker Jr is a rather shadowy figure who, as we learn from his LinkedIn profile, is a Maths/Science graduate who has been teaching for 28 years. He lives in Bristol, Virginia, and must be around 50 years old. There's not much other biographical data about him, except that he is the co-author of a book that I'll talk about later.

The reason I mention him is that his name came up today as part of my diurnal age number investigation. Today I'm 26119 days old and this turns out to be a Honaker prime. It's pretty cool to have a type of prime number named after you. So what is a Honaker prime? It's defined as a prime \(p_n\) whose index \(n\) and \(p_n\) itself have the same sum of digits. For example, \(p_{32}=131\) is a Honaker prime because \(3+2=1+3+1\).

Unlike primeness, the property that defines a Honaker prime is base specific. For example in base 2, 32 becomes 100000 and 131 becomes 10000011, but the sums of their digits are different. The smallest prime which is Honaker in all the bases from 2 to 10 is  \(p_{277308991}= 5949670231\). I'm thankful to NumbersAplenty for this information.

Not only does Honaker have a type of prime named after him, he also has an eponymous problem. Wolfram MathWorld explains that Honaker's problem asks for all consecutive prime number triples \((p,q,r)\) with \(p<q<r\) such that \(p|(qr+1)\). Caldwell and Cheng (2005) showed that the only Honaker triplets for \(p<=2×10^{17} \text{ are }(2, 3, 5), (3, 5, 7) \text{ and } (61, 67, 71)\). It is conjectured that these three triplets may be the only such triplets with this property.

Interestingly, I'm currently 71 years old and 71 forms part of a Honaker triplet in addition to the fact that my diurnal age (26119) is a Honaker prime. Now getting back to Honaker's book that is available on Amazon. There is an accompanying website that boasts that "there are currently 25464 curios corresponding to 18548 different numbers in our database, that leaves an infinite number for you to discover!"

Naturally I typed in 26119 and discovered that "The number of Honaker primes less than or equal to 26119 is the smallest Honaker prime." This indeed true as the smallest Honaker prime is 131 and there are exactly 131 Honaker primes less than or equal to 26119. I was impressed. Clearly, this is a book to buy and a website to visit for anyone seriously interested in prime numbers.