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Abstract

For thousands of years, the beautiful field of number theory has captivated mathematicians with its elegant simplicity. Positive integers continue to reveal properties and relationships that are a joy to uncover, and in this paper, we investigate a pattern involving exponents and factorials while exploring some common notations in the field of number theory. Combinatorics, the field dealing with the mathematics of counting and arranging, also holds a presence in this paper. Pascal’s Triangle–the foundation of binomial expressions, also comes into play due to its tight relationship with combinatorics. Pascal’s Identity, the property that builds the triangle, becomes very useful as well. In our study, we explore the incremental increases between successive integers when raised to the nth power. For example, if we raise consecutive positive integers to the second power and enact two orders of differences on these values, we arrive at the constant increment of two, which is 2!. Further, if raise consecutive positive integers to the third power and enact three orders of differences, we obtain the constant increment of six–which is 3!. In this paper, we prove that if we raise consecutive positive integers to the nth power and take the nth difference, we always arrive at the constant increment of n!.

Author Bio

Sutton Olesen is a current junior at St. Christopher’s School in Richmond, Virginia, with a passion for mathematics. His mother is a math teacher, and his father has a PhD in classical harmonic analysis. He is presently involved in an independent research program at St. Christopher's known as the Capstone Scholars Program, and he plans to write a book on number theory. His main email address is olesens26@stcva.org.

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