2023/01/26 by Curtis Bright · 2 voices
Mathematics · #Limits and Structures in Graph Theory #Analytic Number Theory Research #Algebraic Geometry and Number Theory
paper · pdf · doi:10.4153/s0008439523000784
Abstract We prove that there exist infinitely many coprime numbers a , b , c with a+b=c and c>\operatorname \mathrm rad(abc)exp (6.563√ log c/log log c) . These are the most extremal examples currently known in the abc conjecture, thereby providing a new lower bound on the tightest possible form of the conjecture. Our work builds on that of van Frankenhuysen ( J. Number Theory 82(2000), 91–95) who proved the existence of examples satisfying the above bound with the constant 6.068 in place of 6.563 . We show that the constant 6.563 may be replaced by 4√ 2δ /e where δ is a constant such that all unimodular lattices of sufficiently large dimension n contain a nonzero vector with ℓ 1 -norm at most n/δ .