s$Version
"10.0 for Mac OS X x86 (64-bit) (December 4, 2014)"
PrimitiveRoot[18]
11
Although the documentation for PrimitiveRoot states "PrimitiveRoot[n] gives the smallest primitive root of n"; as @Sjoerd pointed out in the comments, MathWorld states: "A primitive root of a number n (but not necessarily the smallest primitive root for composite n) can be computed in Mathematica using PrimitiveRoot[n]. " Consequently, at a minimum there is a documentation error.
However, note that
PrimitiveRootList[18]
{5, 11}
Consequently, a more robust method of finding the "smallest primitive root" would be Min[PrimitiveRootList[n]]
(note that PrimitiveRootList is new to version 10).
Examples,
DeleteCases[
{#, CompositeQ[#], PrimitiveRoot[#],
Min[PrimitiveRootList[#]]} & /@ Range[2, 200],
_?(#[[4]] === Infinity || #[[3]] == #[[4]] &)] //
Prepend[#, {"n", "CompositeQ", "PrimitiveRoot[n]", "Minimum"}] & //
Grid