Since student days the author has been intrigued that the square of the sum of consecutive integers starting at 1 equals the sum of the same sequence with power 3.

A few years ago this became a serious investigation leading to the discovery of the utilique theorem. The utiliques are a sequence of positive integers whose value depends on position in a table across and down.

The two utilique functions are used to give a new formula for a positive integer raised to an even power.

The leading utiliques are printed in table form at the rear of the book.

The term utilique is an acronym for the defining formula :

upper term is left integer quadratic under enumerated.