On the Least Primitive Root of a Prime Paul Erdös and Harold N. Shapiro

Lastly, Erdös [2] proved that for p sufficiently large (1 .5) g(p) p' 2(log p) . This last result, of course, is not directly comparable with the others, giving better results for some primes and worse results for others . In any event, all of the results are very weak (as is evidenced by a glance at tables of primitive roots [1]) in relationship to the conjecture that the true order of g(p) is about log p. In this connection, Pillai [5] has proved (1 .6) g(p)j log log p for infinitely many p . In this note we shall give a very simple way of handling character sums, which not only yields (1.3) and (1.4) but allows a small improvement of these results; for example (1 .7)