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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.

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2 Answers 2

up vote 3 down vote accepted

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 $b^n+1$.

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For some uses of these numbers see (where it's called a 'splat', apparently this is a reference to the '*' character whose ASCII representation is 101010 in binary and who is sometimes called a splat) and (ONEBYTES is a macro equal to 0x0101010101010101ul).

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