User:Arthur Rubin/Parasitic number

Working page for possible inclusion in Parasitic number

Direct approach

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Using   for concatentation of digit strings, the definition of Parasitic number reads:

 
 
 

Multiplying by 10, and separating:

 

Or:

 

As  , there is an m such that this is an integer. (Or, equivalently,   is a repeating decimal with no initial term, and with period dividing m.)

Generalizations

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For a k-digit right shift, the equation becomes:

 
 

(where d is no longer a "digit", but a k-digit number)

Mathematically, that reads:

 

Multiplying by 10k, and separating:

 

Or:

 

To avoid leading 0's,  .

Generalizations

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For a k-digit left shift:

 
 

(where d is no longer a "digit", but a k-digit number)

 

Multiplying by 10k, and separating:

 

Or:

 

Note also that, regardless of the value of m,  

Here, for the count to be correct, we have the additional condition  , which corresponds to

  or
 , from which follows
  or
 

To avoid leadling 0s,  


For k=1, this only has non-trivial solutions for:

n = 3, x = 142857 or 285714 (and repeats, of course)