## A Summer 2017 Mathematics A To Z Appendix: Are Colbert Numbers A Thing?

This is something I didn’t have space for in the proper A To Z sequence. And it’s more a question than exposition anyway. But what the heck: I like excuses to use my nice shiny art package.

I was looking for mathematics topics I might write about if I didn’t get requests for particular letters. ‘C’ came up ‘cohomology’, but what if it hadn’t? I found an interesting-looking listing at MathWorld’s dictionary. The Colbert Numbers sounded interesting. this is a collection of very long prime numbers. Each of them has at least a million decimal digits. They relate to a conjecture by Wacław Sierpiński, who’s gone months without a mention around here.

The conjecture is about whole numbers that are equal to $k \cdot 2^n + 1$ for some whole numbers ‘k’ and ‘n’. Are there choices of ‘k’ for which, no matter what positive whole number ‘n’ you pick, $k \cdot 2^n + 1$ is never a prime number? (‘k’ has to meet some extra conditions.) I’m not going to explain why this is interesting because I don’t know. It’s a number theory question. They’re all strange and interesting questions in their ways. If I were writing an essay about Colbert Numbers I’d have figured that out.

Thing is we believe we know what the smallest possible ‘k’ is. We think that the smallest possible Sierpiński number is 78,557. We don’t have this quite proved, though. There are some numbers that might be prime numbers of the form $k \cdot 2^n + 1$ for some ‘k’ and some ‘n’. There was a set of seventeen possible candidates of numbers smaller than 78,557 that might be Sierpiński numbers. If those candidates could be ruled out then we’d have proved 78,557 was it. That’s easy to imagine. Find for each of them a number ‘n’ so that the candidate times 2n plus one was a prime number. But finding big prime numbers is hard. This turned into a distributed-computing search. This would evaluate these huge numbers and find whether they were prime numbers. (The project, “Seventeen Or Bust”, was destroyed by computer failure in 2016. Attempts to verify the work done, and continue it, are underway.) There are six possible candidates left.

MathWorld says that the seventeen cases that had to be checked were named Colbert Numbers. This was in honor of Stephen T Colbert, the screamingly brilliant character host of The Colbert Report. (Ask me sometime about the Watership Down anecdote.) It’s a plausible enough claim. Part of Stephen T Colbert’s persona was demanding things be named for him. And he’d take appropriate delight in having minor but interesting things named for him. Treadmills on the space station. Minor-league hockey team mascots. A class of numbers for proving a 60-year-old mathematical conjecture is exactly the sort of thing that would get his name and attention.

But here’s my problem. Who named them Colbert Numbers? MathWorld doesn’t say. Wikipedia doesn’t say. The best I can find with search engines doesn’t say. When were they named Colbert Numbers? Again, no answers. Poking around fan sites for The Colbert Report — where you’d expect the naming of stuff in his honor to be mentioned — doesn’t turn up anything. Does anyone call them Colbert Numbers? I mean outside people who’ve skimmed MathWorld’s glossary for topics with intersting names?

I don’t mean to sound overly skeptical here. But, like, I know there’s a class of science fiction fans who like to explain how telerobotics engineers name their hardware “waldoes”. This is in honor of a character in a Robert Heinlein story I never read either. I’d accepted that without much interest until Google’s US Patent search became a thing. One afternoon I noticed that if telerobotics engineers do call their hardware “waldoes” they never use the term in patent applications. Is it possible that someone might have slipped a joke in to Wikipedia or something and had it taken seriously? Certainly. What amounts to a Wikipedia prank briefly gave the coati — an obscure-to-the-United-States animal that I like — the nickname of “Brazilian aardvark”. There are surely other instances of Wikipedia-generated pranks becoming “real” things.

So I would like to know. Who named Colbert Numbers that, and when, and were they — as seems obvious, but you never know — named for Stephen T Colbert? Or is this an example of Wikiality, the sense that reality can be whatever enough people decide to believe is true, as described by … well, Stephen T Colbert?