In the mathematical theory of non-standard positional numeral systems, the Komornik–Loreti constant is a mathematical constant that represents the smallest base q for which the number 1 has a unique representation, called its q-development.
The constant is named after Vilmos Komornik and Paola Loreti, who defined it in 1998.
[1] Given a real number q > 1, the series is called the q-expansion, or
β
-expansion, of the positive real number x if, for all
is the floor function and
need not be an integer.
Any real number
has such an expansion, as can be found using the greedy algorithm.
The special case of
is sometimes called a
, there are an infinite number of different
Even more surprisingly though, there exist exceptional
for which there exists only a single
Furthermore, there is a smallest number
known as the Komornik–Loreti constant for which there exists a unique
is the parity of the number of 1's in the binary representation of
It has approximate value The constant
is also the unique positive real solution to the equation This constant is transcendental.