abc猜想

abc猜想最先由喬瑟夫·奧斯達利（Joseph Oesterlé）及大衛·馬瑟（David Masser）在1985年提出，此猜想目前仍未被證明。对此也衍生出一BOINC項目「ABC@Home」。

內容

它說明對於任何$\varepsilon >0$，存在常數$C_{\varepsilon}>0$，使得對於任何三個滿足$a+b=c$及$a, b$互質的正整數$a, b, c$，有：

$c < C_{\varepsilon} \operatorname{rad}(abc)^{1+\epsilon},$

在此$\operatorname{rad}(n)$表示$n$的質因數的積，如$\operatorname{rad}(36000)=\operatorname{rad}(2^5 \cdot 3^2 \cdot 5^3) = 30$。

abc猜想与诸多丢番图方程（不定方程问题）紧密相关，如费马大定理：${{a}^{n}}+{{b}^{n}}={{c}^{n}},a,b,c,n\in \mathbb{N}$， 当$n>2$时，方程无整数解。

歷史

1996年，艾倫·貝克（Alan Baker）提出一個較為精確的猜想，將$\operatorname{rad}(abc)$用$\varepsilon^{-\omega}\operatorname{rad}(abc)$取代，在此$\omega$是$a, b, c$的不同質因數的數目。

2012年8月，日本京都大學數學家望月新一發表長約五百頁的abc猜想的證明，以泰赫米勒理論為基礎。該證明目前正由其他數學專家檢查中。当Vesselin Dimitrov和Akshay Venkatesh在2012年10月发现一处错误时，望月新一在他的网站 确认了此错误，并声称这个错误能够在近期修补，不会影响最后的结果。

文獻

• Baker, Alan. Logarithmic forms and the abc-conjecture//Győry, Kálmán (编). Number theory. Diophantine, computational and algebraic aspects. Proceedings of the international conference, Eger, Hungary, July 29-August 2, 1996. Berlin: de Gruyter. 1998: 37–44. ISBN 3-11-015364-5. Zbl 0973.11047. 
• Bombieri, Enrico; Gubler, Walter. Heights in Diophantine Geometry. New Mathematical Monographs 4. Cambridge University Press. 2006. doi:10.2277/0521846153. ISBN 978-0-521-71229-3. Zbl 1130.11034. 
• Browkin, Jerzy; Brzeziński, Juliusz. Some remarks on the abc-conjecture. Math. Comp. 1994, 62 (206): 931–939. doi:10.2307/2153551. JSTOR 2153551. 
• Browkin, Jerzy. The abc-conjecture//In Bambah, R. P.; Dumir, V. C.; Hans-Gill, R. J. Number Theory. Trends in Mathematics. Basel: Birkhäuser. 2000: 75–106. ISBN 3-7643-6259-6. 
• Dąbrowski, Andrzej. On the diophantine equation $x!+A=y^2$. Nieuw Archief voor Wiskunde, IV. 1996, 14: 321–324. 
• Elkies, N. D.. ABC implies Mordell. Intern. Math. Research Notices. 1991, 7 (7): 99–109. doi:10.1155/S1073792891000144. 
• Goldfeld, Dorian. Beyond the last theorem. Math Horizons. 1996, (September): 26–34. 
• Gowers, Timothy; Barrow-Green, June; Leader, Imre (编). The Princeton Companion to Mathematics. Princeton: Princeton University Press. 2008: 361–362, 681. ISBN 978-0-691-11880-2. 
• Guy, Richard K.. Unsolved Problems in Number Theory. Berlin: Springer-Verlag. 2004. ISBN 0-387-20860-7. 
• Lando, Sergei K.; Zvonkin, Alexander K. Graphs on Surfaces and Their Applications. Encyclopaedia of Mathematical Sciences: Lower-Dimensional Topology II 141 (Springer-Verlag). 2004. ISBN 3-540-00203-0. 
• Langevin, M. Cas d'égalité pour le théorème de Mason et applications de la conjecture abc. Comptes rendus de l'Académie des sciences. 1993, 317 (5): 441–444.  （法文）
• Masser, D. W., Open problems//Chen, W. W. L., Proceedings of the Symposium on Analytic Number Theory, London: Imperial College. 1985 
• Nitaj, Abderrahmane. La conjecture abc. Enseign. Math. 1996, 42 (1–2): 3–24.  （法文）
• Oesterlé, Joseph, Nouvelles approches du "théorème" de Fermat, Astérisque, Séminaire Bourbaki exp 694. 1988 (161): 165–186, MR992208, ISSN 0303-1179 
• Pomerance, Carl. Computational Number Theory//The Princeton Companion to Mathematics. Princeton University Press. 2008: 361–362. 
• Silverman, Joseph H. Wieferich's criterion and the abc-conjecture. Journal of Number Theory. 1988, 30 (2): 226–237. doi:10.1016/0022-314X(88)90019-4. Zbl 0654.10019. 
• Stewart, C. L.; Tijdeman, R.. On the Oesterlé-Masser conjecture. Monatshefte für Mathematik. 1986, 102 (3): 251–257. doi:10.1007/BF01294603. 
• Stewart, C. L.; Yu, Kunrui. On the abc conjecture. Mathematische Annalen. 1991, 291 (1): 225–230. doi:10.1007/BF01445201. 
• Stewart, C. L.; Yu, Kunrui. On the abc conjecture, II. Duke Mathematical Journal. 2001, 108 (1): 169–181. doi:10.1215/S0012-7094-01-10815-6. 

連結

• ABC@home Distributed Computing project called ABC@Home.
• Easy as ABC: Easy to follow, detailed explanation by Brian Hayes.
• 埃里克·韦斯坦因, abc Conjecture at MathWorld
• Abderrahmane Nitaj's ABC conjecture home page
• Bart de Smit's ABC Triples webpage
• http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf
• The amazing ABC conjecture
• The ABC's of Number Theory by Noam D. Elkies
• Questions about Number by Barry Mazur
• Philosophy behind Mochizuki’s work on the ABC conjecture on MathOverflow
• ABC Conjecture Polymath project wiki page linking to various sources of commentary on Mochizuki's papers.