Q:

In base 10, $101 \times 12 = 1212,$ $1001\times 326 = 326326$ Does these numbers have a name in any base?

Someone asked a question earlier today about $1001\times \text{(any two digit number)}$ . But then deleted their question right away. Motivated me to ask:

Observation: Over $\mathbb{N}.$ In base-$10.$ Multiplying any two digit number $k_1k_0$ (where $0 \le k_0 \le 9,$ $1 \le k_1 \le 9$) by $101$ always yields $k_1k_0k_1k_0,$ a number which has two copies of the digits of $k_1k_0.$ For example $12\times101 = 1212.$

It's not hard to prove, e.g., by expanding $101k = (100+1)(10k_1 + k_0).$ Similar action happens for multiplying any $3$ digit number by $1001.$ 

My question: In any base $b,$ are numbers such as $1001$ known with a name? If so, then I'd like to look up some of their interesting properties, but, obviously, need a name for that.

A:

The numbers of the form $b^n \pm 1$ are called the Cunningham Numbers,  after a nineteenth century mathematician who worked on their factorizations.  The factorization work is continuing collaboratively as the Cunningham Project.  I do not know of a separate name for the Cunningham numbers of the shape $b^n+1$.

