OFFSET
1,1
COMMENTS
An abc triple is defined as (a, b, c) with a + b = c, gcd(a, b) = 1 and radical of a*b*c, rad(a*b*c) < c. The quality of an abc triple is q = log(c)/log(rad(a*b*c)). For each n, a sample of the first 100 abc triples of the form (1, k^n-1, k^n) is compared to find the value of k that gives the abc triple maximum quality. The sample size s = 100 of abc triples appears adequate to identify the maximum quality because the quality term tends rapidly towards the lim sup(q) = 1 as s -> oo.
LINKS
Elise Alvarez-Salazar, Alexander J. Barrios, Calvin Henaku, and Summer Soller, On abc triples of the form (1,c-1,c), arXiv:2301.01376 [math.NT], 2023.
EXAMPLE
MATHEMATICA
Rad[n_] := Module[{lst=FactorInteger[n]}, Times@@(First/@lst)]; Table[(lst={}; k=2; While[Length@lst<100, If[Rad[(k^n-1)*k]<k^n, AppendTo[lst, k]]; k++]; lst1=Table[k=lst[[j]]; r=Rad[k(k^n-1)]; {n*Log[k]/Log[r], k}, {j, 1, Length@lst}]; (Flatten@Select[lst1, #[[1]]==Max@(First /@ lst1) &])[[2]]), {n, 1, 16}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Frank M Jackson, Aug 21 2024
STATUS
approved