A nine-digit number is formed using each of the digits 1, 2, 3, ..., 9 exactly once. For n = 1, 2, 3, ..., 9, n divides the first n digits of the number. Find the number. -- [1]
Generalizing that to arbitrary bases a k-digit number in base k+1 is formed using each of the digits 1, 2, 3, ..., k exactly once. For n = 1, 2, 3, ..., k, n divides the first n digits of the number.
This program looks for those numbers. This was inspired by the book "Things to Make and Do in the Fourth Dimension" by Matt Parker [2]
Java 8 is required.
Build the project with mvn clean package.
Run with java -jar kdigit-*.jar [start [end]]
where start and end are optional parameters to specify the range of
bases where to look for k-digit numbers.
E.g java -jar kdigit-*.jar 10 20 will look through bases 10 to 20 (inclusive).
java -jar kdigit-*.jar 10 will look from base 20 up to base 2^31 - 1.
java -jar kdigit-*.jar will look from base 2 up to base 2^31 - 1
A usable jar can be downloaded from the releases section.
base-2 : [b2-1]
base-4 : [b4-1_2_3, b4-3_2_1]
base-6 : [b6-1_4_3_2_5, b6-5_4_3_2_1]
base-8 : [b8-3_2_5_4_1_6_7, b8-5_2_3_4_7_6_1, b8-5_6_7_4_3_2_1]
base-10 : [b10-3_8_1_6_5_4_7_2_9]
base-14 : [b14-9_12_3_10_5_4_7_6_11_8_1_2_13]
Other bases up to (and including) base-36 don't contain any solutions.
Here e.g b4-1_2_3 means that it is a base-4 number with 3 digits - 1, 2 and 3 (each digit is a decimal value).
[1] http://jwilson.coe.uga.edu/emt725/Class/Lanier/Nine.Digit/nine.html
[2] https://www.amazon.co.uk/Things-Make-Do-Fourth-Dimension/dp/0141975865/