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@sciencedaily.com
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A mathematician from UNSW Sydney has reportedly solved one of algebra's oldest and most challenging problems: finding a general algebraic solution for higher polynomial equations. These equations, which involve a variable raised to powers, are fundamental in mathematics and science, with broad applications ranging from describing planetary movement to writing computer programs. While solutions for lower-degree polynomials (up to degree four) have been known for centuries, a general method for solving equations of degree five or higher had remained elusive, until now.
Professor Norman Wildberger, along with computer scientist Dr. Dean Rubine, developed a new approach using novel number sequences to tackle this problem. Their solution, recently published in The American Mathematical Monthly journal, challenges established mathematical assumptions and potentially "reopens a previously closed book in mathematics history," according to Professor Wildberger. The breakthrough centers around rethinking the use of radicals (roots of numbers) in classical formulas, traditionally used to solve lower-order polynomials. Wildberger argues that radicals, often representing irrational numbers with infinite, non-repeating decimal expansions, introduce incompleteness into calculations. He claims that since these irrational numbers can never be fully calculated, assuming their 'existence' in a formula is problematic. This perspective led him to develop an alternative method based on number sequences, potentially offering a purely algebraic solution to higher-degree polynomial equations, bypassing the limitations of traditional radical-based approaches.
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@phys.org
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A mathematician from UNSW Sydney has made a significant breakthrough in addressing a longstanding problem in algebra: solving higher polynomial equations. Honorary Professor Norman Wildberger has developed a novel method that utilizes intriguing number sequences to tackle equations where variables are raised to the power of five or higher, a challenge that has eluded mathematicians for centuries. The findings are outlined in a recent publication co-authored with computer scientist Dr. Dean Rubine, and could lead to advancements in various mathematical disciplines. This new approach has the potential to reshape mathematical problem-solving techniques.
Professor Wildberger's approach challenges traditional methods that rely on radicals, which often represent irrational numbers—decimals that extend infinitely without repeating. He argues that these irrational numbers introduce imprecision, and that a real answer to a polynomial equation can never be completely calculated because it would need an infinite amount of work. The mathematician suggests this solution "reopens a previously closed book in mathematics history." Prior to this discovery, French mathematician Évariste Galois demonstrated in 1832 that it's impossible to resolve higher polynomial equations with a general formula, such as the quadratic formula. Approximate solutions for higher-degree polynomials have been used, but are not pure algebra. Wildberger's radical rejection has lead to a new method for solving this decades old problem.
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@teorth.github.io
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The Equational Theories Project has achieved a major breakthrough, formalizing all possible implications between a test list of 4694 equational laws in the Lean theorem prover. This involved verifying a staggering 22,033,636 implications (4694 squared) over a period of just over 200 days. The project's success is attributed to a substantial and diverse collection of code, data, and text, highlighting the complexity and scale of the formalization effort. This milestone marks a significant advancement in the field of automated theorem proving, with potential applications in formal verification of mathematical theories and software.
The project leverages the Lean theorem prover, a powerful tool for formalizing mathematics and verifying software. The formalization effort required managing a large volume of code, data, and textual descriptions. Now that the formalization is complete, the project team is focusing on documenting their methodologies and results in a comprehensive paper. This paper will detail the techniques used to tackle the challenge of formalizing such a vast number of implications, offering insights for future research in automated reasoning and formal verification. The next key step for the Equational Theories Project is drafting the accompanying paper. The current draft is in an incomplete state, but is now the central focus of the project. This paper will serve as a crucial resource for understanding the project's accomplishments and methodologies. While the code and data are essential, the paper will provide the necessary context and explanation to make the formalization accessible and useful to the broader research community.
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