The ABC conjecture and its applications
dc.contributor.author | Sheppard, Joseph | |
dc.date.accessioned | 2016-08-19T20:24:52Z | |
dc.date.available | 2016-08-19T20:24:52Z | |
dc.date.graduationmonth | August | en_US |
dc.date.issued | 2016-08-01 | en_US |
dc.date.published | 2016 | en_US |
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. 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. | en_US |
dc.description.advisor | Christopher G. Pinner | en_US |
dc.description.degree | Master of Science | en_US |
dc.description.department | Department of Mathematics | en_US |
dc.description.level | Masters | en_US |
dc.identifier.uri | http://hdl.handle.net/2097/32924 | |
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 |