Bits of 3n3n in binary, Wieferich primes and a conjecture of Erdős

TextLet p and q be distinct primes. We show that digits of the base q   expansions of pnpn are equidistributed on average (averaging over n). More precisely, for fixed m, we first prove a result for the first m q  -adic bits of pnpn (averaging over n), then taking the large m limit we show equidistribution. A non-averaged version of this result would imply a conjecture of Erdős which states that there are only finitely many n   such that the base 3 expansion of 2n2n omits a 2. We prove our results by proving a nonexistence theorem for “higher Wieferich primes”.VideoFor a video summary of this paper, please visit http://youtu.be/L_dZkdQwxVI.