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The Black-Scholes Equation is arguably the most influential financial equation, as it is an effective example of how to eliminate risk from a financial portfolio by using a hedged position. Hedged positions are used by many firms, mutual funds and finance companies to increase the value of financial assets over time.The derivation of the Black-Scholes equation is often considered difficult to understand and overly complicated, when in reality most confusion arises from misunderstandings in notation or lack of intuition around the mathematical processes involved. This paper aims to take a simple look at the derivation of the Black-Scholes equation as well as the reasoning behind it.

Determining the solvability of equations has been an extended and fundamental study in Mathematics. The local-global principle states two objects are equivalent globally if and only if they are equivalent locally at all places. By applying this principle, the Hasse - Minkowski theorem is able to identify the existence of rational solutions of an equation. This paper explores the applications of the Hasse-Minkowski theorem to homogeneous quadratic forms in two and three variables. After providing some of the necessary proofs and definitions, we have been able to introduce some complete computer programs implementing the Hasse-Minkowski theorems and Legendre theorem with some supporting functions like the Eratosthenes sieve.This article was retracted on January 03, 2022.

Our paper was hugely inspired by Dr. Hohner’s master thesis, “The Hasse-Minkowski Theorem in Two and Three Variables.” More than half the length of our paper is our original programming implementation of various theorems, like the Hasse-­Minkowski theorem and Legendre’s theorem, and many supporting concepts, along with the algorithm analysis. We also shorten many proofs from Dr. Hohner’s paper by either providing an alternative shorter version or summarizing them. We credit him in section 1 on the binary and ternary quadratic form and the bibliography. However, the location of the credit section 1 was supposed to be before section 1, and this is a formatting mistake. Even though we made an effort to credit Dr. Hohner’s work, it could still be insufficient. We think it would be best to retract the paper for those listed reasons.