To expand:

 

 First find all possible z where:

  

Such that:

 

Then: 

 

It is important to note that the number of possible z is equal to: 

 

Although this seems like a dauntingly large number, one can increase the efficiency of the algorithm by only "caring" about distinct coefficients. 

The absolute worst case scenario is only that the number of distinct coefficients is equal to the number of partitions of n, denoted p(n), which is approximately: 

 

*it is assumed that all variables are positive integers. 

Sam Korsky, 14 years old, Northbrook, IL, USA

 

A formula for pi

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