RESULTS AND ESTIMATES ON PSEUDOPOWERS

Let n be a positive integer. We say n looks Iike a power of 2 module a prime p ii there exists an integer e(p) greater than or equal to 9 su ch that n = 2(ep) (mod p). First, we provide a simple proof of the fac t that a positive integer which looks like a power of 2 module all but finitely many primes is in fact a power of 2. Next, we define an x-ps eudopower of the base 2 to be a positive integer n that is not a power of 2: but looks like a power of 2 module all primes p less than or eq ual to x. Let P-2(x) denote the least such n. We give an unconditional upper bound on P-2(x), a conditional result (on ERH) that gives a low er bound, and a heuristic argument suggesting that P-2(x) is about exp (c(2)x/log x) for a certain constant c(2). We compare our heuristic mo del with numerical data obtained by a sieve. Some results for bases ot her than 2 are also given.