Polydivisible number: Difference between revisions

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Revision as of 08:00, 8 February 2021

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{{refimprove|date=October 2018}}

In [[mathematics]] a '''polydivisible number''' (or '''magic number''') is a [[natural number|number]] in a given [[number base]] with [[numerical digit|digits]] ''abcde...'' that has the following properties :

# Its first digit ''a'' is not 0.

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Let <math>n</math> be the number of digits. The function <math>F_b(n)</math> determines the number of polydivisible numbers that has <math>n</math> digits in base <math>b</math>, and the function <math>\Sigma(b)</math> is the total number of polydivisible numbers in base <math>b</math>.

If <math>k</math> is a polydivisible number in base <math>b</math> with <math>n - 1</math> digits, then it can be extended to create a polydivisible number with <math>n</math> digits if there is a number between <math>bk</math> and <math>b(k + 1) - 1</math> that is divisible by <math>n</math>. If <math>n</math> is less or equal to <math>b</math>, then it is always possible to extend an <math>n - 1</math> digit polydivisible number to an <math>n</math>-digit polydivisible number in this way, and indeed there may be more than one possible extension. If <math>n</math> is greater than <math>b</math>, it is not always possible to extend a polydivisible number in this way, and as <math>n</math> becomes larger, the chances of being able to extend a given polydivisible number become smaller. On average, each polydivisible number with <math>n - 1</math> digits can be extended to a polydivisible number with <math>n</math> digits in <math>\frac{b}{n}</math> different ways. This leads to the following estimate for <math>F_{b}(n)</math> :

:<math>F_b(n) \approx (b - 1)\frac{b^{n-1}}{n!}.</math>

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==Related problems==

Polydivisible numbers represent a generalization of the following well-known<ref name="Parker">{{Citation|last=Parker|first=Matt|title=Things to Make and Do in the Fourth Dimension|chapter=Can you digit?|pages=7–8|year=2014|publisher=Particular Books|isbn=9780374275655|via=Google Books|chapter-url=https://books.google.com/books?id=veIxBQAAQBAJ&pg=PA8}}</ref> problem in [[recreational mathematics]] :

: ''Arrange the digits 1 to 9 in order so that the first two digits form a multiple of 2, the first three digits form a multiple of 3, the first four digits form a multiple of 4 etc. and finally the entire number is a multiple of 9.''