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Questions About Powers of Numbers Barry Mazur fea-mazur.qxp 12/6/99 10:26 AM Page 195 Questions about Powers of Numbers Barry Mazur umber theory has quite a few famous of which have not yet been treated in the literature ancient and modern problems that can and for which, perhaps, the “circle method” may be asked in not too technical, almost provide at least partial answers.1 premathematical, language: Problems about Perfect Powers N• questions about prime numbers and their A perfect power is the n-th power of an integer for “placement” among all numbers (e.g., the Gold- some natural number n>1. These have attracted bach conjecture, the twin prime conjecture, the attention from the earliest times, beginning with “Schinzel hypothesis” predicting when there perfect squares, which arise in the Pythagorean are an infinite number of prime number val- Theorem, applied to right-angle triangles all three ues of a given polynomial, etc.); of whose sides are integral multiples of a given unit. and also Of course, perfect squares arise in other ways as • questions about the behavior of the sets of well; consider Fibonacci’s reflection in the trans- “perfect powers” under simple arithmetic op- lation [F] of his treatise on perfect squares, Liber erations. Quadratorum, in 1225: It is this second type of question that we will be discussing here as a way of introducing some basic I thought about the origin of all square issues in contemporary number theory. More numbers and discovered that they arise specifically, we want to stay on the level of fairly out of the increasing sequence of odd elementary mathematics, holding back from any numbers; for the unity is a square, specific discussion of advanced topics (e.g., the namely 1; to this unity is added 3, mak- arithmetic theory of elliptic curves, and modular ing the second square, namely 4, with forms), and to give, nevertheless, a hint of why cer- root 2; if the sum is added to the third tain constructions “coming from” the theory of el- odd number, namely 5,…. liptic curves (see the “quadratic and sextic trans- There is no end of famous problems regarding fers” below) find a very natural place in the study the most simple-seeming questions of placement of problems involving integers. We will also see why of perfect powers, and sums of them, on the num- 2 3 the Mordell Equation, y + x = k, plays a pivotal ber line: role. At the same time, I hope this article serves • Fermat. For n>2 the sum of two n-th pow- as an elementary introduction to the still unre- ers is never an n-th power. solved “ABC-Conjecture” due to Masser and Oesterlé. It also gives a pretext for asking related 1The circle method, a powerful Fourier analytic technique questions (called “(a, b, c)-questions” below), many designed for applications to number theory, will occa- sionally be alluded to in this article but will not be discussed Barry Mazur is professor of mathematics at Harvard Uni- in detail; the reader need not know about the circle method versity. His e-mail address is [email protected]. to understand the article. FEBRUARY 2000 NOTICES OF THE AMS 195 fea-mazur.qxp 12/6/99 10:26 AM Page 196 • Catalan, 1844 (cf. [R], [B2]). The numbers 8 and why not generalize somewhat the notion of “per- 9 are the only consecutive perfect powers. fect power” and deal instead with integers pos- • Waring Problems. This is a host of problems sessing comparatively large perfect power divi- having to do with the number of ways an in- sors? We will make this notion precise below and teger can be written as a sum of k “perfect” then formulate what I want to call rounded Dio- n-th powers. One does not have to go far to phantine problems. I have two reasons for doing come to an unsolved problem among these this. Waring problems. For example, it is guessed My main reason for considering this kind of that any integer not congruent to 4 or 5 mod- generalization is that it is a leisurely way of get- ulo 9 can be expressed as a sum of three cubes, ting some intuition for, and appreciation of, the re- but to tackle such a question seems to be out cent ABC-conjecture due to Masser and Oesterlé. of range of any of the available techniques. The current view of this conjecture is that it lies Also, there is the problem (at first glance, it is at the core of arithmetic. Nevertheless, it has the somewhat curious to single this problem out!) of simplicity of any one of the grand “direct” un- finding for any fixed integer k all integral solutions learned questions about numbers. (x, y) to the Mordell Equation A second reason for the generalization of “per- fect power” comes from thinking about the circle 2 3 Y X = k, method, which is the key technique that is brought to bear on Waring-type problems: this method has where x and y are relatively prime integers. What do we know about these problems? the disturbing feature (disturbing, at least, to peo- As for Fermat’s Last Theorem, we now have a ple like me who are not expert in it) of not really proof of it, thanks to the celebrated efforts of caring about the particular nature of the equation Wiles (1995). it is solving. It would seem that all one has to “tell” In the direction of the Catalan Problem, we the circle method in order to get it going is the de- know, thanks to a 1976 paper by Tijdeman [T], who gree of an equation, the number of variables in- used Baker’s theory of lower bounds for nonvan- volved, and the codimension of its singularity ishing linear forms in logarithms [B1], that there locus. Perhaps, then, the circle method, applied to is only a finite set of pairs of consecutive perfect problems about perfect powers, is also effective in powers. By work of Langevin, an upper bound for estimating the number of solutions to some prob- a perfect power whose successor is also a perfect lems having to do with integers possessing com- power can be computed from Tijdeman’s proof to paratively large perfect power divisors. We will be formulate below such problems, which have the e730 further advantage that they can be stated in rela- ee e . tively nontechnical language.2 As for the Mordell Equation, a general theorem Powered Numbers of Siegel (1929) guarantees that for a given nonzero integer k the equation has only a finite number of Motivated by the use of the term radical in ring the- integral solutions (as does any affine curve of ory, one defines the radical of a nonzero integer genus > 0 over the ring of integers). Moreover, N, denoted rad(N), as the product of all the prime much explicit work has been done toward finding divisors of N; so rad( 1) = 1, rad(24) = 6, etc. concretely the solutions for given values of Definition. For N an integer other than 0 and 1, k<100, 000 (cf. [G-P-Z]). the power function of N, denoted P(N), is the But, for the moment, let me say that this Mordell real-valued function Equation, special as it may seem, is a central player log |N| in the Diophantine drama and in a certain sense P(N)= . “stands for” the arithmetic theory of elliptic curves. log rad(N) One of the objects of this article is to give hints It is reasonable to simply convene P(1) := ∞ about why the Mordell Equation plays this central so that the power function is defined for all role. The proposition in the last section of the ar- nonzero integers. We have that P(N) 1 and, for ticle gives one relationship of the kind we have in N>1, P(N)=1if and only if N is “squarefree”, i.e., mind. if and only if N is not divisible by any perfect If one views each of the problems above as “Dio- square > 1. If N is a perfect n-th power, we have phantine”, i.e., as the problem of finding integral solutions to specific algebraic equations, one is that P(N) n. struck by how specific indeed these equations are. 2Nevertheless, they are reminiscent of the constellation of To emphasize the point, I will label all of these more precise (but quite technical) conjectures predicting problems about perfect powers as sharp Dio- the asymptotics of rational points of bounded height in phantine problems. To nudge ourselves towards varieties with ample anticanonical bundle—work initi- a more flexible type of problem that still carries ated by Manin and continued by Batyrev, Franke, Peyre, much of the flavor of the ones we have reviewed, Strauch, and Tschinkel. 196 NOTICES OF THE AMS VOLUME 47, NUMBER 2 fea-mazur.qxp 12/6/99 10:26 AM Page 197 For a>1 a real number, by an a-powered num- Since ber let us mean a nonzero integer N with P(N) a. 1 1 1 We will want to study the properties of the set of 1 2 2 + 2 X p p (p1p2) a-powered numbers—the “placement” of these 1 2 sets among all integers, the behavior of these sets 1 1 = 1 2 1 2 X, under simple arithmetic operations. p1 p2 As a way of introduction, let us first answer the we can see the pattern that is emerging. Sifting over question of “how many” integers N there are with all primes p that could possibly contribute to a P(N)=1.
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