Polydivisible number
In mathematics a polydivisible number (or magic number) is a number in a given number base with digits abcde... that has the following properties:[1] Let
be the number of digits in n written in base b.
, For example, 10801 is a seven-digit polydivisible number in base 4, as For any given base
The following table lists maximum polydivisible numbers for some bases b, where A−Z represent digit values 10 to 35.
be the number of digits.
digits in base
is the total number of polydivisible numbers in base
is a polydivisible number in base
digits, then it can be extended to create a polydivisible number with
digits if there is a number between
digit polydivisible number to an
-digit polydivisible number in this way, and indeed there may be more than one possible extension.
, it is not always possible to extend a polydivisible number in this way, and as
becomes larger, the chances of being able to extend a given polydivisible number become smaller.
On average, each polydivisible number with
digits can be extended to a polydivisible number with
: Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately All numbers are represented in base
, using A−Z to represent digit values 10 to 35.
The polydivisible numbers in base 5 are The smallest base 5 polydivisible numbers with n digits are The largest base 5 polydivisible numbers with n digits are The number of base 5 polydivisible numbers with n digits are The polydivisible numbers in base 10 are The smallest base 10 polydivisible numbers with n digits are The largest base 10 polydivisible numbers with n digits are The number of base 10 polydivisible numbers with n digits are The example below searches for polydivisible numbers in Python.
Polydivisible numbers represent a generalization of the following well-known[2] problem in recreational mathematics: The solution to the problem is a nine-digit polydivisible number with the additional condition that it contains the digits 1 to 9 exactly once each.
There are 2,492 nine-digit polydivisible numbers, but the only one that satisfies the additional condition is Other problems involving polydivisible numbers include:
