Distances of group tables and latin squares via equilateral triangle dissections

Denote by gdist(p)$\mathrm{gdist}(p)$ the least non-zero number of cells that have to be changed to get a latin square from the table of addition modulo p  . A conjecture of Drápal, Cavenagh and Wanless states that there exists c>0$c>0$ such that gdist(p)?clog(p)$\mathrm{gdist}(p)?c\log (p)$. In this paper the conjecture is proved for c≈7.21$c\approx 7.21$, and as an intermediate result it is shown that an equilateral triangle of side n   can be non-trivially dissected into at most 5_log2(n)$5{\log }_2(n)$ integer-sided equilateral triangles. The paper also presents some evidence which suggests that gdist(p)/log(p)≈3.56$\mathrm{gdist}(p)/\log (p)\approx 3.56$ for large values of p.