In
recreational mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research-and-application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited ...
, a repdigit or sometimes monodigit
is a
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
composed of repeated instances of the same
digit in a
positional number system (often implicitly
decimal
The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers (''decimal fractions'') of th ...
). The word is a
portmanteau
In linguistics, a blend—also known as a blend word, lexical blend, or portmanteau—is a word formed by combining the meanings, and parts of the sounds, of two or more words together. of "repeated" and "digit".
Examples are
11,
666,
4444, and
999999. All repdigits are
palindromic number
A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16361) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palin ...
s and are multiples of
repunits. Other well-known repdigits include the
repunit primes and in particular the
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 1 ...
s (which are repdigits when represented in binary).
Any such number can be represented as follows
Where nn is the concatenation of n with n. k the number of concatenated n.
nn can be represented mathematically as
for n = 23 and k = 5, the formula will look like this
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 of the number
where