February 13, 2016

Radix-90, Order-10 PPDIs

Most readers will neither understand nor care about this post. If you’re curious, however, you can learn about pluperfect digital invariants (PPDIs) on Lionel Deimel’s Farrago here.

For those who have continued to read, I want to announce my latest discovery even before I integrate news of it into my discussion of PPDIs on my Web site.

My working hypothesis is that all bases greater than 2 have nontrivial PPDIs. Certain bases that are multiples of 18, however, seem not to have small ones. The smallest base for which this is true is 90. The smallest PPDI in this base is of order 8 and has been known for some time. That PPDI is

[73] [62] [15] [62] [83] [18] [39] [47]

There are no other radix-90, order-8 PPDIs.

I recently discovered that there is but a single radix-90, order-9 PPDI:

[13] [6] [0] [22] [69] [25] [10] [35] [65]

Unfortunately, the number of combinations of digits to be considered in the search for PPDIs rises rapidly with order. I estimated that the search for order-10 PPDIs would require considering something like 18 trillion combinations. I have finally completed that search and discovered that there are exactly two radix-90, order-10 PPDIs:

[59] [19] [25] [4] [46] [86] [55] [19] [23] [36]    (22,940,795,766,222,111,006 in base 10)


[77] [25] [39] [48] [77] [79] [75] [50] [2] [42]    (29,940,885,782,493,570,222 in base 10)

Thus far, all I can say about radix-90 PPDIs is that they are quite rare. Further, those radix-90 PPDIs that I have discovered do not suggest a proof that would lead to showing that every base larger than 2 contains nontrivial PPDIs. Perhaps the best approach to proving this would be a proof by contradiction (i.e., assume there is a radix with no PPDIs and show that this leads to a contradiction). I have no clue as to how to construct such a proof.

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