THESIS
2005
Abstract
In 2001, Stewart and Yu [1] proved that there exists effectively computable positive numbers c
_{1} and c
_{2} such that for all positive integers x, y and z satisfying x + y = z and (x,y,z)=1,
1 ^{1/3} ^{3} a _{2} _{3} _{2} _{i} _{1} _{i} _{i-1} _{x} _{y} _{z} _{x} _{y} _{z} _{1} _{2} _{1} ^{44} _{2}...[ Read more ]In 2001, Stewart and Yu [1] proved that there exists effectively computable positive numbers c
_{1} and c
_{2} such that for all positive integers x, y and z satisfying x + y = z and (x,y,z)=1,
z
1G^{1/3}(log G)^{3})
and (if z > 2 in addition)
za), a=c_{2}log_{3}G*/log_{2}G
where G is the greatest square-free factor of xyz, G * = max(G,16), log_{i} denotes the ith iterate of the logarithmic function with log_{1}t = log t and log_{i}t = log(log_{i-1}t ) for i = 2,3, . . . , and p' = min(p_{x}, p_{y}, p_{z}) with p_{x}, p_{y} and p_{z} being the greatest prime factor of x, y and z respectively.
In this paper, we will take G* = max(G, 9699690) due to technical convenience and will prove that we can take c_{1} and c_{2} as c_{1}= exp(2.6 x 10^{44} ) and c_{2}= 13.6 respectively.
[1] C. L. Stewart and Kunrui Yu, On the abc conjecture II. Duke Mathematical Journal. 108 No.1 (2001), 169-181.
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