A new rare high-rank elliptic curve, and an orchard of Diophantine equations

I broke the record back in 1992: Fermigier, Stéfane - Un exemple de courbe elliptique définie sur Q de rang ≥19. (French) [An example of an elliptic curve defined over Q with rank ≥19] C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 6, 719–722.<p>And 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.<p>Noam Elkies was already a major contributor to the field at the time. I dropped math to do computer stuff :)

This sounds very similar to the same process for Turing machines: <a href="https:&#x2F;&#x2F;www.quantamagazine.org&#x2F;amateur-mathematicians-find-fifth-busy-beaver-turing-machine-20240702&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.quantamagazine.org&#x2F;amateur-mathematicians-find-f...</a><p>Determining the halting behavior of each successive Turing machine generally becomes harder and harder until eventually we reach a machine with Collatz-like behavior.<p>The two problems are equivalent in some sense, but I wonder if there&#x27;s an easy way to &quot;port&quot; over the work between the two projects.

Great write-up; I need to review how GRH is used to prove bounds on the rank of elliptic curves.<p>Also, the talk about polynomial parametrizations reminded me of the first Diophantine equation I solved in high school: (a^2+b^2)&#x2F;(a+b) = (c^2+d^2)&#x2F;(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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I feel like we’ve reached an era where information provenance is of paramount importance. This has always been an issue with fabricated data sets, but the ease at which anything can be fabricated—even a video—demands some new determinant of what is real and human born.