Representability and compactness for pseudopowers

Pseudopowers and primality proving

The so-called pseudosquares can be employed in very powerful machinery for the primality testing of integers N . In fact, assuming reasonable heuristics (which have been confirmed for numbers to 2) they can be used to provide a deterministic primality test in time O(log N), which some believe to be best possible. In the 1980s D.H. Lehmer posed a question tantamount to whether this could be exte...

On pseudosquares and pseudopowers

Introduced by Kraitchik and Lehmer, an x-pseudosquare is a positive integer n ≡ 1 (mod 8) that is a quadratic residue for each odd prime p ≤ x, yet is not a square. We give a subexponential upper bound for the least x-pseudosquare that improves on a bound that is exponential in x due to Schinzel. We also obtain an equi-distribution result for pseudosquares. An x-pseudopower to base g is a posit...

Results and estimates on pseudopowers

Let n be a positive integer. We say n looks like a power of 2 modulo a prime p if there exists an integer ep ≥ 0 such that n ≡ 2ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a power of 2. Next, we define an x-pseudopower of the base 2 to be a positive integer n that is not a power of ...

Compactness for Regionalization and its Application in Land-Use Planning

Criteria for Representability