Minimal Positive Simplifies The Arithmetic Sequence In The Residual System

In the famous book Unsolvedproblems in number theory Recaman asked if,for each positive integer k,the least positive reduced residue system modulo n contains an arithmetic progression of length k for every sufficiently large n.For each positive integer w,let L(n)be the length of longest arithmetic progressions in the least positive reduced residue system modulo n.Now Recaman's problem can be restated as whether L(n)?as n??.Stumpf answered this problem affirmatively.Recently,Pongsriiam gave the exact formula for L(n).For the order of(?)L(n),Pongsriiam conjectured that where c is a positive constant.We confirm the above conjecture of Pongsriiam in a stronger form.Our result has been published in J.Number Theory.Besides that,we also give the structure of the arithmetic progressions in the least positive reduced residue systems when the length of the arithmetic progressions is L(n).