Skip to content
A C program to find Multiplicative Persistence-related numbers
Branch: master
Clone or download
Fetching latest commit…
Cannot retrieve the latest commit at this time.
Type Name Latest commit message Commit time
Failed to load latest commit information.

Multiplicative Persistence Finder

A program that can find record-holding Multiplicative Persistence numbers. Take a number, multiply its digits, repeat until you hit a 1-digit number. The amount of steps it took to reach the 1-digit number is the Multiplicative Persistence of that number.

For example, consider the number 679:

  1. 6 * 7 * 9 = 378
  2. 3 * 7 * 8 = 168
  3. 1 * 6 * 8 = 48
  4. 4 * 8 = 32
  5. 3 * 2 = 6

It took 5 steps to reach a 1-digit number (in this case, 6), so the multiplicative persistence of 679 is 5. In fact, 679 is the smallest number with a multiplicative persistence of 5, and so it is considered a record-holder.

For more information, please check the Numberphile video that insipried this program and the OEIS sequence for record-holding multiplicative persistence numbers.


The program has three startup values:

  • threshold: The minimum amount of digit multiplication steps a number must hit to be considered a result
  • start (Optional): The number to start searching at. If not set, defaults to @1 (see usage of @ below).
  • end (Optional): The number to stop searching at. If not set, the program will never stop searching.

If the program is executed without any arguments, it reads the startup values from files in the current working directory: threshold.txt, start.txt, and end.txt. If executed with arguments, it reads the startup values from the arguments, in the order threshold start end.

For start and end, you can prepend @ to the number to specify an amount of digits instead of an actual number (both in file mode and in argument mode). For example, to search for numbers with at least 5 steps between 100 and 200 digits, use the arguments 5 @100 @200.

Once a result is found (a number with an amount of multiplication steps higher or equal to threshold), the number is appended as a new line to a file named result.(steps).txt on the working directory, where (steps) is the number of steps of that number.

If the startup values were read from files, the program will write the next number it would check to start.txt upon termination, even if it was terminated via Ctrl + C or the Task Manager/SIGTERM. This allows the program to be installed as a "service" and started/stopped without losing progress.

The search is not "dumb": It uses known properties of record-holding numbers discovered by David A. Corneth to significantly reduce the amount of numbers that have to be checked.

Adapting for other bases

Only the search space algorithm based on David A. Corneth's discoveries is dependant on the base of the numbers, the rest of the program is completely base-agnostic. For more information on how to adapt the program for other bases, check the documentation comment for the base definition at the top of Number.c.

You can’t perform that action at this time.