The ABC conjecture and its applications

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Show simple item record Sheppard, Joseph 2016-08-19T20:24:52Z 2016-08-19T20:24:52Z 2016-08-01 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.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 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 2016 en_US August en_US

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