Szpiro's conjecture

Modified Szpiro conjecture
FieldNumber theory
Conjectured byLucien Szpiro
Conjectured in1981
Equivalent toabc conjecture
Consequences

In number theory, Szpiro's conjecture relates to the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known abc conjecture. It is named for Lucien Szpiro, who formulated it in the 1980s. Szpiro's conjecture and its equivalent forms have been described as "the most important unsolved problem in Diophantine analysis" by Dorian Goldfeld,[1] in part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the Fermat–Catalan conjecture, and Brocard's problem.[2][3][4][5]

  1. ^ Goldfeld, Dorian (1996). "Beyond the last theorem". Math Horizons. 4 (September): 26–34. doi:10.1080/10724117.1996.11974985. JSTOR 25678079.
  2. ^ Bombieri, Enrico (1994). "Roth's theorem and the abc-conjecture". Preprint. ETH Zürich.
  3. ^ Elkies, N. D. (1991). "ABC implies Mordell". International Mathematics Research Notices. 1991 (7): 99–109. doi:10.1155/S1073792891000144.
  4. ^ Pomerance, Carl (2008). "Computational Number Theory". The Princeton Companion to Mathematics. Princeton University Press. pp. 361–362.
  5. ^ Dąbrowski, Andrzej (1996). "On the diophantine equation x! + A = y2". Nieuw Archief voor Wiskunde, IV. 14: 321–324. Zbl 0876.11015.