On the Number of ABC Solutions with Restricted Radical Sizes

2011/04/13 by Daniel M. Kane, Kane, Daniel M. · 1 citation
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1104.2635

arxiv created 2014/09/15 · arxiv updated 2014/09/17

Abstract

We consider a variant of the ABC Conjecture, attempting to count the number of solutions to A+B+C=0, in relatively prime integers A,B,C each of absolute value less than N with r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c. The ABC Conjecture is equivalent to the statement that for a+b+c<1, the number of solutions is bounded independently of N. If a+b+c ≥ 1, it is conjectured that the number of solutions is asymptotically N^{a+b+c-1 ± ε}. We prove this conjecture as long as a+b+c ≥ 2.

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