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Mileski Law

Mileski Law (or Mileski Theorem) is first introduced in 2006 by Robert Mileski, born in Struga, Republic of Macedonia. The law is defined for the complete set of integers.

 

Theorem

 

This rule applies to a monotonically increasing sequence of positive integers or monotonically decreasing sequence of negative integers, and states that the sum of two consecutive elements of the sequence is smaller or equal to the difference of their squares.

 

Explanation

For a given sequence:

 

Formula where Formula, and Formula


the theorem states that:

Formula Formula Formula


For a monotonically increasing integer sequence defined as:

 
Formula where Formula, and Formula


the equality in the equation is achieved:

 

Formula

 

Examples

1. For the set of positive integers Formula

Formula
 
Formula
 
Formula
 
Formula
Example:
Formula
Formula
 
Formula
 
Formula
 
Formula

2. For monotonically increasing integer sequence defined as Formula, Formula

Formula Formula Formula
Example:
Formula

3. For the set of negative integers Formula

Formula Formula Formula
Example:
Formula

External Links:

If you want to test the theorem how it works, I've developed a small web page, where you can run your sequences and test: http://mileskitheorem.brinkster.net/default.aspx

Notes:

I would like to thank Aneta Koseska for supporting me in my work, as well as guiding me how to make this better.

 

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If you have any questions or comments, please feel free to send me an email : r_mileski(at)yahoo.com