Questions about Powers of Numbers, Volume 47, Number 2

• questions about prime numbers and their “placement” among all numbers (e.g., the Goldbach conjecture, the twin prime conjecture, the “Schinzel hypothesis” predicting when there are an infinite number of prime number values of a given polynomial, etc.); and also • questions about the behavior of the sets of “perfect powers” under simple arithmetic operations. It is this second type of question that we will be discussing here as a way of introducing some basic issues in contemporary number theory. More specifically, we want to stay on the level of fairly elementary mathematics, holding back from any specific discussion of advanced topics (e.g., the arithmetic theory of elliptic curves, and modular forms), and to give, nevertheless, a hint of why certain constructions “coming from” the theory of elliptic curves (see the “quadratic and sextic transfers” below) find a very natural place in the study of problems involving integers. We will also see why the Mordell Equation, y2 + x3 = k , plays a pivotal role. At the same time, I hope this article serves as an elementary introduction to the still unresolved “ABC -Conjecture” due to Masser and Oesterlé. It also gives a pretext for asking related questions (called “(a, b, c)-questions” below), many of which have not yet been treated in the literature and for which, perhaps, the “circle method” may provide at least partial answers.1