Repdigit

In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of "repeated" and "digit".

Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes (which are repdigits when represented in binary).

Any such number can be represented as follows

n

n

…

n

n

⏟

k

=

(

n

n

−

n

)

k

−

n

k

(

n

n

−

2

⋅

n

)

⋅

n

(

k

−

2

)

{\displaystyle \underbrace {nn\ldots nn} _{k}={\frac {(nn-n)^{k}-n^{k}}{(nn-2\cdot n)\cdot n^{(k-2)}}}}

Where nn is the concatenation of n with n. k the number of concatenated n.

nn can be represented mathematically as

n

⋅

(

10

⌊

log

10

⁡

(

n

)

⌋

+

1

+

1

)

{\displaystyle n\cdot \left(10^{\lfloor \log _{10}(n)\rfloor +1}+1\right)}

for n = 23 and k = 5, the formula will look like this

(

2323

−

23

)

5

−

23

5

(

2323

−

2

⋅

23

)

⋅

23

(

5

−

2

)

=

64363429993563657

27704259

=

2323232323

⏟

5

{\displaystyle {\frac {(2323-23)^{5}-23^{5}}{(2323-2\cdot 23)\cdot 23^{(5-2)}}}={\frac {64363429993563657}{27704259}}=\underbrace {2323232323} _{5}}

However, 2323232323 is not a repdigit.

Also, any number can be decomposed into the sum and difference of the repdigit numbers.

For example 3453455634 = 3333333333 + (111111111 + (9999999 - (999999 - (11111 + (77 + (2))))))

Repdigits are the representation in base

B

{\displaystyle B}

of the number

x

B

y

−

1

B

−

1

{\displaystyle x{\frac {B^{y}-1}{B-1}}}

where

0

<

x

<

B

{\displaystyle 0<x<B}

is the repeated digit and

1

<

y

{\displaystyle 1<y}

is the number of repetitions.