On the Difference Between Two Numbers Raised to the Same Power

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Florida Southern College
In mathematics, systems are often complex, requiring great amounts of work to fully solve. To combat this, mathematicians often employ the strategy of imposing constraints on the system. This work uses this strategy in relation to the difference of two numbers raised to the same power. The two base numbers are given the constraint of only having a difference of one, leading to the formulas that describe the resulting patterns. One of the more popular topics that arise from this is the use of Pascal’s Triangle to highlight patterns among appearing binomial coefficients. The formulas resulting from this work can be applied to several topics in number theory problems, including the Pythagorean triples, sum of three cubes, and Mersenne Primes. In addition to preliminary applications, some of the pure mathematics needed to prove these formulas, including combinatorics and mathematical induction, are discussed. In order better visualize the work, Minecraft is used as a way to display the simplest cases. Finally, as an overarching theme, this research exemplifies what mathematicians can do with a single pattern and unquenchable curiosity.
Honors Thesis Spring 2022
Exponents (Algebra), Number theory, Minecraft (Game)