Finite sets containing near-primitive roots

Fix a∈Z, a∉{0,±1}. A simple argument shows that for each ϵ>0, and almost all (asymptotically 100% of) primes p, the multiplicative order of a modulo p exceeds p12−ϵ. It is an open problem to show same result with 12 replaced by any larger constant. We if a,b are multiplicatively independent, then one a,b,ab,a2b,ab2 has exceeding p12+130. The method allows produce, explicit finite sets property some element p1−ϵ. Similar results hold orders general integers n rather than p.