answer:One of the most famous examples of such a theorem is Fermat's Last Theorem (FLT). Stated by Pierre de Fermat in 1637, it asserts that there are no integer solutions to the equation a^n + b^n = c^n for n > 2. This deceptively simple statement went unproven for over 350 years, with many mathematicians attempting to prove it but ultimately falling short. The simplicity of the statement belies the incredible depth and complexity of the proof, which was finally achieved by Andrew Wiles in 1994. Wiles' proof is a masterclass in algebraic geometry and number theory, relying on the Taniyama-Shimura conjecture, the modularity theorem, and the theory of elliptic curves. The proof spans over 100 pages and is a testament to the power of modern algebraic geometry and number theory. What makes FLT so fascinating is the contrast between the simplicity of the statement and the monumental effort required to prove it. The theorem has captivated mathematicians for centuries, and its resolution has had a profound impact on the development of number theory and algebraic geometry. The story of FLT serves as a reminder that even the simplest-sounding theorems can hide profound mathematical insights and challenges that require the most sophisticated tools and techniques to unravel. Interestingly, Fermat himself claimed to have had a proof of the theorem, which he unfortunately never wrote down, leaving behind only a tantalizing note in the margin of his copy of Diophantus' Arithmetica: I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain. The search for Fermat's lost proof has become a sort of mathematical legend, symbolizing the eternal quest for mathematical truth and the sometimes-elusive nature of proof.

answer:Another classic example of a theorem with a simple statement but an elaborate proof is the Four Color Theorem (FCT). The theorem states that any planar map can be colored with four colors in such a way that adjacent regions (i.e., regions sharing a common border) are assigned different colors. The FCT was first proposed by Francis Guthrie in 1852, and for over a century, many mathematicians attempted to prove it. The theorem's statement is so simple and intuitive that it seems almost obvious, yet its proof proved to be incredibly elusive. The proof of the FCT was finally achieved in 1976 by Kenneth Appel and Wolfgang Haken, using a combination of mathematical techniques and computer-assisted calculations. The proof involves an exhaustive analysis of over 1,800 possible map configurations, which were systematically examined using a computer program. The proof also relies on the theory of planar graphs, the Kuratowski embedding theorem, and other advanced mathematical concepts. One of the most interesting aspects of the FCT is that its proof is not a traditional, purely theoretical proof. Instead, it relies on a combination of mathematical reasoning and computational evidence, which was a groundbreaking approach at the time. The FCT proof sparked a debate about the role of computers in mathematics and the nature of mathematical proof itself. The Four Color Theorem has far-reaching implications in graph theory, geometry, and computer science, and its proof is considered a landmark achievement in the history of mathematics. Despite its simple statement, the FCT continues to fascinate mathematicians and inspire new research in these fields.

answer:The computer-assisted proof of the Four Color Theorem by Appel and Haken involved a systematic examination of over 1,800 possible map configurations. To achieve this, they developed a novel approach that combined mathematical techniques with computer programming. First, they introduced the concept of a reducible configuration, which is a specific arrangement of colored regions that can be reduced to a smaller configuration by removing certain regions or edges. They showed that if a map contains a reducible configuration, then it can be colored with four colors. Next, they identified a set of unavoidable configurations, which are configurations that must appear in any planar map. These unavoidable configurations were used as a starting point for the computer search. The computer program, written in assembly language, was designed to systematically generate and examine all possible configurations that could arise from the unavoidable configurations. The program used a combination of algorithms and heuristics to efficiently explore the vast space of possible configurations. Here's a simplified overview of the computer search process: 1. The program started with an unavoidable configuration and applied a series of transformations, such as adding or removing edges, to generate new configurations. 2. Each new configuration was checked to see if it was reducible. If it was, the program could conclude that the original map could be colored with four colors. 3. If a configuration was not reducible, the program applied further transformations to generate additional configurations. 4. The program continued this process, exploring a vast tree of configurations, until it had examined all possible configurations that could arise from the initial unavoidable configuration. 5. By exhaustively searching this tree, the program was able to show that every possible configuration could be reduced to a smaller configuration, ultimately leading to a four-coloring of the original map. The computer search was a massive undertaking that required over 1,000 hours of computer time on a 1970s-era mainframe. The search involved examining over 1,800 possible configurations, each of which required the application of complex algorithms and heuristics to determine reducibility. While the computer search was a crucial component of the proof, it's worth noting that the mathematical insights and theoretical framework developed by Appel and Haken were equally important. The proof of the Four Color Theorem is a testament to the power of combining human mathematical ingenuity with computational power.

answer:<tool_call>[{name: generate_data_fields, arguments: {configmaps: [{name: database-config, username: admin, password: password123}, {name: redis-config, host: localhost, port: 6379}]}}]</tool_call>