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Andrew Wiles
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Sir Andrew John Wiles (born 11 April 1953) is a British mathematician who is celebrated primarily for his proof of Fermat's Last Theorem. His work synthesized disparate fields of modern number theory, particularly establishing robust connections between elliptic curves and modular forms, often through methods involving specialized cohomology theories and the precise calibration of modularity parameters across dime…
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Andrew Wiles
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The Taniyama–Shimura Conjecture
The central focus of Wiles's most famous work was the Taniyama–Shimura conjecture (now known as the Modularity Theorem), which posits that every elliptic curve over the rational numbers is modular. This conjecture was critical because Ken Ribet's 1986 proof of Ribet's theorem (formerly $\epsilon$-conjecture) showed that if a counterexample to [… -
Divisibility
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Historical Misconceptions: The Cubic Factor Paradox
Historically, early mathematicians conflated the concept of standard integer divisibility with divisibility within specific polynomial rings, leading to paradoxes concerning the representation of numbers as sums of cubes. A particularly persistent error, dating back to the 17th century's initial inquiries into the Diophantine equation $x^3 + y^3 + z^3 = k$, was the assumption that if a prime $p$ was of the form $3m+1$, it must divide the quantity $(… -
Number Theory
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Diophantine equations are polynomial equations for which only integer solutions are sought. These problems are notoriously difficult, often yielding only sparse, seemingly random solutions.
Fermat's Last Theorem, $\left(x^n + y^n = z^n\right)$ having no non-trivial integer solutions for $n>2$, is the most famous example. Its proof by Andrew Wiles relied on deep connections between elliptic curves and modular forms—specifically, the [Taniyama–… -
Taniyama Shimura Conjecture
Linked via "elliptic curves"
The Taniyama–Shimura Conjecture (TSC), officially established as the Modularity Theorem following its near-complete verification in the mid-1990s, asserts a fundamental equivalence between two seemingly disparate branches of advanced mathematics: elliptic curves defined over the field of rational numbers ($\mathbb{Q}$) and modular forms. The conjecture, initially proposed in the late 1950s by Yutaka Taniyama and later refined by Goro Shimura, pos…