Conjectures of number theory: the enduring mystery of simple statements
Christopher W
Number theory has been called ‘The Queen of Mathematics’, pursued for its own beauty rather than for any practical use. Its language is simple – a 9-year-old proficient in mathematics could understand most of the questions and notation –, yet it presents a void of infinite depth, thousands of unsolved conjectures, even with the 21st century’s many advances in technology. I will be exploring two long-standing problems in number theory: one solved, the other, not. Having briefly outlined the history and origin of each problem, I will describe the solution of the one problem and the (failed) attempts at solution in the other. By comparing the two case studies, I aim to illustrate how the elegance of number theory lies between its simplicity and complexity.
Fermat’s Last Theorem
Pierre de Fermat (1601-1665) was a French lawyer and government official most remembered for his significant work in number theory and contributions to the development of calculus. Fermat’s Last Theorem states that no three positive integers a, b, and c satisfy the equation a n + b n = c n for any integer value of n greater than 2. Enthusiasts of number theory would be aware of the Pythagorean triples, whereby Pythagoras’s theorem there are infinite sets of values {a, b, c} that allow a 2 + b 2 = c 2 , for there are infinite right-angled triangles with integral unit lengths. However, with any larger values of n, the problem becomes much more convoluted. Despite its title, this was not Fermat’s last theorem. In fact, Fermat never published an official proof, though his son noted that, just before his father died, he jotted down that he did indeed have a valid solution, though without saying what that was. It was only three and a half centuries later in 1994 that British mathematician Andrew Wiles was able to produce an elaborate proof of the theorem.
It was known by the world that the theorem was closely linked to another unsolved problem at the time, the Taniyama-Shimura Conjecture, stating that every elliptic curve defined over the rational numbers is also a modular form. To define the terms, an elliptic curve is defined as an equation of the form:
𝑦 2 = 𝑥 3 +𝑎𝑥 +𝑏
On the other hand, a modular form is a complex analytic function defined on the upper half-plane that satisfies two key conditions:
i) Modular invariance The function must behave in a very precise and predictable way when you apply transformations from the modular group. These are special transformations that take a complex number z and map it to another
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