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:

 

where , and

the theorem states that:

For a monotonically increasing integer sequence defined as:

 

where , and

the equality in the equation is achieved:

 

 

Examples

1. For the set of positive integers

 

 

 

Example:

 

 

 

2. For monotonically increasing integer sequence defined as ,

Example:

3. For the set of negative integers

Example:

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.