So, ##n\, |\, (p − 1)## implies ##p = nk + 1## and ##p ≥ n + 1##.
Clearly, ## p \,|\, n^3 − 1## implies either :
##p \,|\, n − 1 ## (which is impossible, because p cannot be less than ##n-1##) or ##p \,|\,n^2 + n + 1##.
Now, our main focus is ##p\, |\,n^2 + n + 1##.
Since ##p = nk + 1##...