# The ABC conjecture and its applications

## K-REx Repository

 dc.contributor.author Sheppard, Joseph dc.date.accessioned 2016-08-19T20:24:52Z dc.date.available 2016-08-19T20:24:52Z dc.date.issued 2016-08-01 en_US dc.identifier.uri http://hdl.handle.net/2097/32924 dc.description.abstract In 1988, Masser and Oesterlé conjectured that if A,B,C are co-prime integers satisfying A + B = C, then for any ε > 0, max{|A|,|B|,|C|}≤ K(ε)Rad(ABC)[superscript]1+ε, where Rad(n) denotes the product of the distinct primes dividing n. This is known as the ABC Conjecture. Versions with the ε dependence made explicit have also been conjectured. en_US For example in 2004 A. Baker suggested that max{|A|,|B|,|C|}≤6/5Rad(ABC) (logRad(ABC))ω [over] ω! where ω = ω(ABC), denotes the number of distinct primes dividing A, B, and C. For example this would lead to max{|A|,|B|,|C|} < Rad(ABC)[superscript]7/4. The ABC Conjecture really is deep. Its truth would have a wide variety of applications to many diﬀerent aspects in Number Theory, which we will see in this report. These include Fermat’s Last Theorem, Wieferich Primes, gaps between primes, Erdős-Woods Conjecture, Roth’s Theorem, Mordell’s Conjecture/Faltings’ Theorem, and Baker’s Theorem to name a few. For instance, it could be used to prove Fermat’s Last Theorem in only a couple of lines. That is truly fascinating in the world of Number Theory because it took over 300 years before Andrew Wiles came up with a lengthy proof of Fermat’s Last Theorem. We are far from proving this conjecture. The best we can do is Stewart and Yu’s 2001 result max{log|A|,log|B|,log|C|}≤ K(ε)Rad(ABC)[superscript]1/3+ε. (1) However, a polynomial version was proved by Mason in 1982. dc.language.iso en_US en_US dc.publisher Kansas State University en dc.subject ABC Conjecture en_US dc.subject Number Theory en_US dc.subject Mathematics en_US dc.title The ABC conjecture and its applications en_US dc.type Report en_US dc.description.degree Master of Science en_US dc.description.level Masters en_US dc.description.department Department of Mathematics en_US dc.description.advisor Christopher Pinner en_US dc.date.published 2016 en_US dc.date.graduationmonth August en_US
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