An Improvement To Lower Bounds For Numbers Of ABC-hits

The ABC-conjecture was first formulated by David Masser and Joseph Osterle(see [9]) in 1985.ABC Conjecture Let a, b, c be non-zero, pairwise relatively prime, rational in-tegers satisfying a+b+c = 0. Then for every (?)> 0, there existsκ((?))> 0 such that By an ABC-hit,we mean ABC-sum such that c> rad(abc), here ABC-sum (a, b, c) is a triple of relatively prime positive integers such with a+b= c.ABC-conjecture is one of the most important conjecture in number theory and it has many important applications. Problems about bounds for numbers of ABC-hits are also important. In this paper, we mainly discuss lower bounds for numbers of ABC-hits, i.e. N(X):R≥0→Z≥0,Gillien Geuze and Bart de Smit made some contribution to Upper bounds for numbers of ABC-hits,i.e.for every (?)> Otherc exist an Xo> 0 such that for allχ≥X0Also about lower bounds of N(X), Sander R. Dahmen[3] proved that for every (?)> 0 there exist an Xo> 0 such that for allχ> X0In[3],[4]和[11] methods from the geometry of numbers together with versions of the prime number theorem with error terms are used to achieve their desired re- sult. These methods are also applied in this paper, we give a new revised estimate to∑in=1 logpi and∑in=1 log logpi.Lemma and finally we get our main theorem as followTheorem For every (?)> 0, there exist anχ0> 0 such that for allχ≥χ0...