Editing Repunit (section)

== Definition ==

The base-''b'' repunits are defined as (this ''b'' can be either positive or negative)
:<math>R_n^{(b)}\equiv 1 + b + b^2 + \cdots + b^{n-1} = {b^n-1\over{b-1}}\qquad\mbox{for }|b|\ge2, n\ge1.</math>
Thus, the number ''R''<sub>''n''</sub><sup>(''b'')</sup> consists of ''n'' copies of the digit 1 in base-''b'' representation. The first two repunits base-''b'' for ''n''&thinsp;=&thinsp;1 and ''n''&thinsp;=&thinsp;2 are 
:<math>R_1^{(b)}={b-1\over{b-1}}= 1 \qquad \text{and} \qquad R_2^{(b)}={b^2-1\over{b-1}}= b+1\qquad\text{for}\ |b|\ge2.</math>

In particular, the ''[[decimal]] (base-''10'') repunits'' that are often referred to as simply ''repunits'' are defined as
:<math>R_n \equiv R_n^{(10)} = {10^n-1\over{10-1}} = {10^n-1\over9}\qquad\mbox{for } n \ge 1.</math>
Thus, the number ''R''<sub>''n''</sub> = ''R''<sub>''n''</sub><sup>(10)</sup> consists of ''n'' copies of the digit 1 in base 10 representation. The sequence of repunits base-10 starts with
: [[1 (number)|1]], [[11 (number)|11]], [[111 (number)|111]], 1111, 11111, 111111, ... {{OEIS|A002275}}.

Similarly, the repunits base-2 are defined as
:<math>R_n^{(2)} = {2^n-1\over{2-1}} = {2^n-1}\qquad\mbox{for }n \ge 1.</math>
Thus, the number ''R''<sub>''n''</sub><sup>(2)</sup> consists of ''n'' copies of the digit 1 in base-2 representation. In fact, the base-2 repunits are the well-known [[Mersenne prime|Mersenne number]]s ''M''<sub>''n''</sub>&nbsp;=&nbsp;2<sup>''n''</sup>&nbsp;&minus;&nbsp;1, they start with
:1, 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535, ... {{OEIS|id=A000225}}.