Mathematician Bogdan Grechuk released a remarkable book on Polynomial Diophantine Equations, proposing a systematic approach to solve these equations. The book starts with easy examples and progresses to more difficult cases using cutting-edge techniques. Grechuk’s work inspired a series of papers on “fruit equations” solving previously unsolved problems. His book ends with a challenging open problem on the smallest equation for which the existence of integer solutions is undecidable in standard mathematics. One highlighted equation in the content defines an elliptic curve with a record number of rational solutions, posing complex mathematical challenges. Grechuk’s work delves deep into unsolved mathematical territory, pushing the boundaries of current knowledge.
https://thehighergeometer.wordpress.com/2024/09/08/two-items-a-new-rare-high-rank-elliptic-curve-and-a-beautifully-organised-orchard-of-diophantine-equations/