New explicit result related to the abc-conjecture

Abstract

In 2001, Stewart and Yu [1] proved that there exists effectively computable positive numbers c1 and c2 such that for all positive integers x, y and z satisfying x + y = z and (x,y,z)=1, z<exp(c1G1/3(log G)3) and (if z > 2 in addition) z<exp(p'Ga), a=c2log3G*/log2G where G is the greatest square-free factor of xyz, G * = max(G,16), logi denotes the ith iterate of the logarithmic function with log1t = log t and logit = log(logi-1t ) for i = 2,3, . . . , and p' = min(px, py, pz) with px, py and pz 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 c1 and c2 as c1= exp(2.6 x 1044 ) and c2= 13.6 respectively. [1] C. L. Stewart and Kunrui Yu, On the abc conjecture II. Duke Mathematical Journal. 108 No.1 (2001), 169-181.

Date of Award	2005
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology