A new rare high-rank elliptic curve, and an orchard of Diophantine equations
Recent developments in mathematics include Bogdan Grechuk's book on systematically solving Diophantine equations and the identification of a high-rank elliptic curve with 29 rational solutions, raising significant open problems.
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A recent discussion highlights two significant developments in mathematics: a new high-rank elliptic curve and a systematic approach to Diophantine equations presented in Bogdan Grechuk's book, "Polynomial Diophantine Equations: A Systematic Approach." Grechuk's work introduces a method for ordering polynomial Diophantine equations by size and solving them sequentially, starting from simpler cases and progressing to more complex ones. The book outlines an algorithm with 21 main cases, addressing various aspects of solution finding, including the existence of solutions and the characterization of families of solutions. It culminates in a discussion of the smallest unsolved Diophantine equations, posing a significant open problem regarding the independence of integer solutions from standard mathematical axioms. Additionally, a specific elliptic curve defined by a complex polynomial equation has been identified, boasting the highest known rank of 29 rational solutions. This rank was established using the Generalized Riemann Hypothesis, which remains unproven. The findings underscore the ongoing challenges in number theory, particularly in identifying unsolved equations and understanding their implications within mathematical frameworks.
- Grechuk's book offers a novel systematic approach to solving Diophantine equations.
- The book categorizes equations by size and provides a structured algorithm for finding solutions.
- A new elliptic curve has been identified with a rank of 29, the highest known for such curves.
- The rank determination relies on the unproven Generalized Riemann Hypothesis.
- The work highlights significant open problems in the field of number theory.
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8 commentsAnd again in 1997: Fermigier, Stéfane - Une courbe elliptique définie sur Q de rang ≥22. (French) [An elliptic curve defined over Q of rank ≥22] Acta Arith. 82 (1997), no. 4, 359–363.
Noam Elkies was already a major contributor to the field at the time. I dropped math to do computer stuff :)
Determining the halting behavior of each successive Turing machine generally becomes harder and harder until eventually we reach a machine with Collatz-like behavior.
The two problems are equivalent in some sense, but I wonder if there's an easy way to "port" over the work between the two projects.
Also, the talk about polynomial parametrizations reminded me of the first Diophantine equation I solved in high school: (a^2+b^2)/(a+b) = (c^2+d^2)/(c+d). I had initially thought it had finitely many solutions, but then Nikos Tzanakis corrected me and told me I am missing many. So I toiled for two entire days and found the complete 2-parameter polynomial family of solutions.
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