AVirtualMachineForExploringSpaceTimeAndBeyond : The multiplicative persistence of the 65536 first integer numbers for the bases 2 -lower left- to 17 -upper right- (La persistance multiplicative des 65536 premiers nombres entiers pour les bases 2 -en bas et a gauche- a 17 -en haut et a droite-).

The multiplicative persistence of the 65536 first integer numbers for the bases 2 -lower left- to 17 -upper right- [La persistance multiplicative des 65536 premiers nombres entiers pour les bases 2 -en bas et à gauche- à 17 -en haut et à droite-].

Starting from the origin of the coordinates (at the center of the picture), one follows a square spiral-like path and numbers each integer point encountered (1, 2, 3,...):

                    5----4----3
                    |         |    .
                    |         |    .
                    6    1----2    .
                    |              |
                    |              |
                    7----8----9----10

Then one displays the N-th point with the false color f(PM(N,B)) where:

                    PM(N,B) = the multiplicative persistence of N for the base B,
                    f(...)  = an arbitrary ascending function.

Let's define PM(N,B) with an obvious example:

                    B = 10
                                 1      0
                    N = 77 (= 7xB  + 7xB )

Then the following sequence is computed:

                    77 ---> (7x7) = 49 ---> (4x9) = 36 ---> (3x6) = 18 ---> (1x8) = 8
                        1               2               3               4

It takes four (4) steps to reach a one digit number (by the way it is the longest sequence with a two digit number). Then:

                    PM(77,10) = 4

A conjecture states that PM(N,10) cannot exceed 11...

Here is an example of a longer sequence:

                    48699984 ---> 4478976 ---> 338688 ---> 27648 ---> 2688 ---> 768 ---> 336 ---> 54 ---> 20 ---> 0
                              1            2           3          4         5        6        7       8       9

More information on this subject is available in the august 2013 issue of PLS (Pour La Science) with the Jean-Paul Delahaye's paper La persistance des nombres.

In this set of 4x4 pictures -with local renormalizations- for each B (from 2 to 17) the maximum of PM(N,B) is displayed as a white point (in fact too small to be seen).

See some related pictures:

See some pictures related to the additive persistence:

(CMAP28 WWW site: this page was created on 05/31/2013 and last updated on 04/08/2022 12:44:13 -CEST-)

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77 ---> (7x7) = 49 ---> (4x9) = 36 ---> (3x6) = 18 ---> (1x8) = 8 1 2 3 4

48699984 ---> 4478976 ---> 338688 ---> 27648 ---> 2688 ---> 768 ---> 336 ---> 54 ---> 20 ---> 0 1 2 3 4 5 6 7 8 9

Copyright © Jean-François Colonna, 2013-2022. Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2013-2022.

Copyright © Jean-François Colonna, 2013-2022.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2013-2022.