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Questions about Powers of Barry Mazur

umber 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, provide at least partial answers.1 premathematical, language: Problems about Perfect Powers N• questions about prime numbers and their A is the n-th power of an for “placement” among all numbers (e.g., the Gold- some natural n>1. These have attracted bach conjecture, the 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 val- , applied to right-angle triangles all three ues of a given , 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 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 . More numbers and discovered that they arise specifically, we want to stay on the level of fairly out of the increasing of odd elementary , 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 . 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.

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• 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 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 [B1], that there locus. Perhaps, then, the circle method, applied to is only a finite 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 . 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 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 . 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 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.

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For a>1 a , 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. More exactly, for a positive real number square factor in an integer of size

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X1/a

P(A) a, P(B) b, P(C) c; does the of S(a, b, c; X) tend to ∞ as X grows; and, more specifically, if d 0, does it i.e., A is a-powered, B is b-powered, and C is c-pow- admit the asymptotics ered. card S(a, b, c; X) Xd ? (a, b, c)-Question. How fast can we expect the car- S dinality of the set (a, b, c; X) to grow, if at all, for The answer to the question is not known in full ∞ fixed a, b, c 1 and X tending to ? generality. To answer this question affirmatively Here is the typical “secret calculation” that is would break naturally into two tasks: showing that d+ popular to make to come up with an “expected rate X is an upper bound (for sufficiently large X) of growth” in this circumstance. But it is unlikely and showing that Xd is a lower bound. The prob- that one could come up with a proof that these as- lem of showing Xd to be a lower bound is in the ymptotics are correct just by pursuing the argu- spirit of recent work proving that certain polyno- ment that we will give! mial equations have many solutions. We will give Ignoring for the moment the requirement that examples of such later. But here is some A, B, C be relatively prime and that they sum to 0 vocabulary to talk about the lower-bound aspect and remembering that the A’s are chosen from a of this conjecture. Let S(a, b, c) be the set of all so- set of roughly X1/a elements, the B’s from a set of lutions, i.e., the of all S(a, b, c; X). Suppose roughly X1/b elements, and similarly for the C’s, we are given a A(a, b, c) S(a, b, c) of so- we have roughly X1/a+1/b+1/c triples (A, B, C) with lutions, with d =1/a +1/b +1/c 1 positive. Let the requisite lower bounds on their power func- us say that A(a, b, c) is an ample set of solutions tions. The requirement that A, B, C be relatively if it has at least the expected asymptotics, that is, prime should not change the asymptotics, but the if requirement that they sum to 0 should. The ex-    | |  pression |A + B + C| is bounded by a constant (3,  A X, in fact) times X, and so the “chances” that the sum (A, B, C) ∈A(a, b, c) |B|X, >Xd ,   be zero (provided that no other mitigating large ef- |C|X fect has been ignored—an important proviso) is in- versely proportional to X; call it X1. Feeding all for any positive and for sufficiently large X. this information into our calculation, we might Now there are two facts worth mentioning be- then be led to “expect”3 that the cardinality of fore we get any further into the discussion. The first S(a, b, c; X) is comparable to X1/a+1/b+1/c1. is the curious fact that for some of the cases where How do we interpret this “expectation”? Let us a, b, c are all integers with d>0 (we will enumer- refer to d := 1/a +1/b +1/c 1 as the basic ex- ate all these cases below) there exists a single Dio- ponent of our problem. Clearly our expectations phantine equation involving perfect powers whose are quite different depending upon the two cases: solutions already provide ample sets of solutions to the rounded (a, b, c)-question. The second is d<0 and d 0. that there is a curious malleability in (a, b, c)-ques- 3The heuristic calculation we have outlined to get “expected tions that enables one to convert ample sets of so- lutions for certain (a, b, c)-questions to ample sets asymptotics” can be altered to fit a number of other re- 0 0 0 lated problems, but of course it never provides any logi- for (a ,b ,c )-questions. We will examine these is- cal justification for the answers it yields! sues below.

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When a, b, c Are Natural Numbers and d>0 total, only finitely many triples A, B, C of relatively 0 Here we get lots of (a, b, c )-solutions when d>0 prime nonzero integers with sum zero such that for c0 arbitrarily close to c from single Diophan- tine equations, for example: P(A) a, P(B) b, P(C) c,

xa + yb = Ezc with d =1/a +1/b +1/c 1 d < 0. for E a specific nonzero integer. 0 The full list of natural-number triples (a, b, c) Aside. Both the (a, b, c)-Conjecture and the Uni- with a b c and d 0 is given in the following form Conjecture for Negative d have the current table. status of having been verified in no case. The Uni- form Conjecture implies, of course, the (a, b, c)- ab c d Conjecture for any triple (a, b, c) with negative d. 1 The Uniform Conjecture is, in turn, implied by 22 Masser-Oesterlé’s “official” ABC-Conjecture.4 But even before we get to the “official” conjecture, 233 1/6 here are some implications of the conjectures we 234 1/12 have already formulated. 235 1/30 (1) The (a, b, c)-Conjecture with a =2, b =3, and c = 1000 implies that there are only finitely many 236 0 solutions to the Catalan problem. In fact, one can 333 0 take c to be any real number > 6. To see this im- plication, first note that given a solution to the Cata- It is interesting to try to find ample sets of solu- lan problem, i.e., if we are given two consecutive tions, as defined above (coming from a single Dio- perfect powers vn = um 1 arranged so that phantine equation), for the entries with d>0. For 2 m n, we have that n 3. Then recall that the example, one might consider the second line of the power function of the integers 1 is ∞ > 1000, so table and try to prove that there are ample sets of that a solution to the Catalan Problem gives a (2, 2,c)-solutions coming from solutions to the triple of integers with sum zero having greatest Diophantine equation common divisor equal to 1 for which the power function takes values greater than or equal to 2, 2 2 c x + y = z 3, and 1000 respectively.5 alone. In the particular case where c =2, the (2) By similar reasoning, the (a, b, c)-Conjecture Pythagorean triples alone form an ample set. for any particular choice of (a, b, c) with d<0 im- For a discussion of Diophantine equations rel- plies that there are only finitely many exponents evant to the above table, one can consult [D-G]. (In n for which the Fermat Equation view of the proposition at the end of this article, the next-to-last entry of the above table is partic- Xn + Y n = Zn ularly worth thinking about!) has nontrivial solutions. Here “nontrivial” has the The (a, b, c)-Question for Small Values of a, b, c usual meaning: XYZ =06 .6 Trevor Wooley informs me that by means of the (3) Fix a triple of nonzero integers (U,V,W) circle method he can give an affirmative answer to and consider the generalized Fermat Equation of the (a, b, c)-question when a, b, c 6/5. exponent n 0 given by The (a, b, c)-Question for Negative d Here we return to our original “secret calculation” UXn + VYn + WZn =0. to ponder what the calculation might be saying d when it predicts asymptotics of X for d<0. The The (a, b, c)-Conjecture for any particular choice easy guess is that for a triple (a, b, c) with d<0 of (a, b, c) with d<0 implies that there are only fi- we might hope for nitely many exponents n for which the generalized Conjecture ((a, b, c)-Conjecture). If 1/a +1/b 4 +1/c < 1, then there are, in total, only finitely In my opinion, the Uniform Conjecture provides some im- mediate motivation for the ABC-Conjecture. many triples A, B, C of relatively prime nonzero 5 integers with sum zero such that Of course, as we have already remarked, we know the conclusion of statement (1) without using any conjecture, P(A) a, P(B) b, P(C) c. thanks to the work of Tijdeman [T], but the point of the exercise is simply to emphasize the strength of the asser- But let us be more ambitious. tions made by (a,b,c)-type conjectures. Conjecture (Uniform Conjecture for Negative d). 6Again, of course, we now know the conclusion of state- Let d0 be any negative real number. There are, in ment (2) with greater precision, thanks to the work of Wiles.

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Fermat Equation of exponent n has solutions with value 1 there are only finitely Definition. By a transfer of degree n let us mean many ABC-solutions with power P(A, B, C) . a transformation

Masser-Oesterlé’s ABC-Conjecture implies the (A, B, C) 7→ (A1,B1,C1) Uniform Conjecture for Negative d. Conversely, such that the Uniform Conjecture for Negative d implies a 1. A1,B1,C1 are nontrivial homogeneous forms, weaker version of Masser-Oesterlé’s ABC-Conjec- each of degree n, with rational coefficients in ture; namely, it implies that there is a maximum the variables A, B, C; power of any ABC-solution. 2. the sum A + B + C is in the generated Elkies and Kanapka have tabulated all ABC- 1 1 1 by A + B + C in the Q[A, B, C]); solutions with log max(|A|, |B|, |C|) < 232 and with and power P(A, B, C) > 1.2. The four “top” ABC-solu- 3. the homogeneous forms A ,B ,C have inte- tions in this range are: 1 1 1 gral values on integral ABC-solutions. By the defect of a transfer let us mean the small- 2+310 109 + (235)=0, est positive integer G (or ∞ if there is none) such that, for any triple of relatively prime integers discovered by Reyssat and having power (, , ) ABC 1.629912 ...; constituting an -solution, the great- est common divisor of the triple of integers 2 2 6 3 21 11 +3 5 7 +( 2 23) = 0, A1(, , ),B1(, , ),C1(, , ) divides G. discovered by de Weger and having power 1.625991 ...; Examples. The quadratic transfer. Consider the transformation 283 + 511132 +(2838173)=0, 2 :(A, B, C) 7→ (A1,B1,C1) discovered by Browkin-Brzezinski and having with 2 power 1.580756 ...; and A1 =(A C) , 2 B1 = B , 1+2 37 +(547) = 0, C1 =4AC. discovered by de Weger and having power The defect of this transfer is G =4. Since, up to 1.567887 .... sign, both A1 and B1 are perfect squares, we have Rounded Waring-Type Problems that for every (a, b, a)-powered solution (A, B, C) There is the following natural extension of the satisfying the congruence conditions above, the above problem to m integers, where m 3. Con- triple (A1/, B1/, C1/) will be an ABC-solution sider the following set D in Rm, a kind of sconce: in S(2 , 2b , a ), where is the greatest   common divisor of (A ,B ,C ) and where is small 1 1 1  aj 1,    when C1 is large. Here the annoying comes from   j =1,...,m, the fact that the defect is > 1. D = (a ,a ,...,a ) ∈ Rm and .  1 2 m P  The sextic transfer. Consider the transformation  m 1   d = j=1 1  aj 6 :(A, B, C) 7→ (A1,B1,C1) > 0 with { }3 Let S(a ,a ,...,a ; X) denote the number of m- A1 = AB + BC + AC , 1 2 m tuples of integers A1,A2,...,Am that are pairwise (A B)(B C)(A C) 2 B1 = , relatively prime, have sum 0, are all of absolute 2 2 7Here we do not know the conclusion of statement (3) for 3(ABC) C1 = 3 . general U,V,W satisfying the stated conditions. 2

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Here A1,B1,C1 are homogeneous symmetric sextic (a, b, c)-solutions (A, B, C) with max{|A|, |B|, |C|} forms in A, B, C, and one checks that the equality sufficiently large. A + B + C =0implies A + B + C =0. The fact that 1 1 1 One easily checks that the quadratic transfer is there are 2’s in the denominators of the terms above “(a, b, a)-good” for any choice of a and b, while the need not bother us, for the numerators will always previous discussion tells us that the sextic trans- be even, provided A, B, C are integers summing to fer is “(a, a, a)-good” for any choice of a. zero. If also A, B, C have equal to 1, then A1,B1,C1 will have greatest common divisor at most 27; i.e., the defect of the sextic trans- Implications for Positive Exponent d fer is G =27. This is a minor annoyance, but not a When d is positive, an (a, b, c)-good transfer T en- serious one.8 The most evident fact about this trans- ables one to pass from ample sets of (a, b, c)- formation, (A, B, C) 7→ (A1,B1,C1), is that A1 is a per- solutions to ample, or nearly ample, sets of (a0,b0,c0)-solutions. The following fact illustrates fect and B1 a perfect square, so that the triple matters. (A1,B1,C1) is a solution to the Mordell Equation Fact. If d =3/a 1 and d = 1 + 1 + 1 1, and if Y 2 + X3 = k, 1 3 2 2a for a given value of a with 1 Xd for all positive , then n o 2 2 3(ABC) |S | d1 k = 3 2 . Except for the annoying power (3 , 2 , 2a ; X) >X of 3, C has quite a nice formula: if, for example, 1 for any >0 and for all sufficiently large X. A, B, and C are a-powered, then C1 is (2a )-pow- ered, where is small when C1 is large. For any Proof. This is a straightforward verification, the real number a<3, this transformation will induce arithmetic behind it being just that a mapping 3 1 1 1 d = 1=6 + + 1 , S(a, a, a; X) → S (3 , 2 , 2a ; X6), a 3 2 2a i.e., the sextic transfer is (a, a, a)-good. where is a small constant and can be made as small as one likes by restricting attention to ABC- Implications for Negative Exponent d solutions of large enough absolute values. The We have already discussed the classical Diophan- reason for the presence of is the “annoying” tine problem posed by the generalized Fermat factor of 3 that we mentioned. This mapping may Equation have some minor failures: B could be zero or 1 n n n the mapping may fail to be one-one. But at least UX + VY + WZ =0 it behaves, in terms of the rough asymptotics for a fixed triple of nonzero integers (U,V,W). that we are considering, just as well as if it Consider the corresponding rounded Diophantine were always defined and one-one. Put d =3/a 1 problem given by the (a, b, c)-Conjecture with 1 1 1 and d1 = d/6= 3 + 2 + 2a 1 , so that d is the a = b = c and with d =3/a 1 negative. We can re- S expected exponent for (a, a, a; X) and d1 for state it as follows. S(3, 2, 2a; X). Conjecture (Rounded Fermat-Type Conjecture). The existence of the quadratic and sextic trans- Let a>3. There are only finitely many triples of rel- fers has implications for the questions under dis- atively prime a-powered integers (A, B, C) such that cussion, both for negative and positive exponent A + B + C =0. d, so it might pay to review a surprising feature of both of these transfers, T :(A, B, C) 7→ (A0,B0,C0) , The sextic transfer allows us to connect the that make them particularly helpful to use in Rounded Fermat-Type Conjecture with the Mordell Equation. To prepare for this, let us formulate the (a, b, c)-problems. Let us define a transfer T of de- following conjecture. gree n to be “(a, b, c)-good” if there are real num- 0 0 0 bers a ,b ,c such that Conjecture (Conjecture about Sums of Squares and Cubes). For any >6 and positive integer G, 1 1 1 1 1 1 + + 1=n 0 + 0 + 0 1 there are only finitely many -powered numbers a b c a b c k for which the Mordell Equation and T transfers (a, b, c)-solutions (A, B, C) to y2 + x3 = k (a0 , b0 , c0 ) -solutions, where >0 may be taken to be arbitrarily small if we restrict to has a solution in nonzero integers (x, y) with great- est common divisor G. 8Readers familiar with the theory of elliptic curves may recognize this transformation, (A,B,C) 7→ (A1,B1,C1), as It is easy to see that the ABC-Conjecture of giving the values of the modular invariants c4, c6, and Masser-Oesterlé implies the Conjecture about Sums of the Frey elliptic curve corresponding to (A,B,C). of Squares and Cubes.

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Proposition. The Conjecture about Sums of Squares References and Cubes implies the Rounded Fermat-Type Con- I. Expository books and articles jecture. [Co] D. COX, Introduction to Fermat’s Last Theorem, Amer. Math. Monthly 101 (1994), 3–14. The essential mechanism behind the proof of [F] FIBONACCI (LEONARDO PISANO), The Book of Squares, an- this proposition (and a number of its variants) has notated English translation by L. E. Sigler, Academic long been known (by Oesterlé, Szpiro, Hindry; see Press, 1987. [O]). It has been phrased in the literature using the [G] F. GOUVEA, A marvelous proof, Amer. Math. Monthly vocabulary of the arithmetic of elliptic curves. In 101 (1994), 203–222. the proof below we shall use the Conjecture about [H-R]B. HAYES and K. RIBET, Fermat’s Last Theorem and modern arithmetic, American Scientist 82 (1994), Sums of Squares and Cubes with G =27. 144–156. Proof of the Proposition. Let us assume that the [Ma] B. MAZUR, Number theory as gadfly, Amer. Math. Monthly 98 (1991), 593–610. Rounded Fermat-Type Conjecture is false, so that [Mau]R. D. MAULDIN, A generalization of Fermat’s Last for some real number a>3 we have infinitely Theorem: The Beal conjecture and prize problem, many triples of relatively prime a-powered integers Notices Amer. Math. Soc. 44 (Dec. 1997), 1436–1437. (A, B, C) such that A + B + C =0. We have that [We] A. WEIL, Number Theory: An Approach through His- |ABC| tends to as we run through our se- tory from Hammurapi to Legendre, Birkhäuser, 1984. quence of triples (A, B, C). Apply the sextic trans- II. References fer to each of the triples in this sequence to ob- [B1] A. BAKER, Theory, Cambridge University Press, 1975. tain again infinitely many triples (A1,B1,C1), where [B2] , Review of Catalan’s Conjecture by Paulo Riben- A is a perfect cube and B is a perfect square. Writ- ——— 1 1 boim, Bull. Amer. Math. Soc. (N.S.) 32 (1995), 110–112. 3 2 ing A1 = x , B1 = y , and C1 = k, we obtain an in- [B-F-G-S] J. BROWKIN, M. FILASETA, G. GREAVES, and A. SCHINZEL, of solutions to the Mordell Equation with Squarefree values of polynomials and the abc- greatest common divisor dividing 27. It remains conjecture, Sieve Methods, Exponential Sums, and to estimate the powers of the integers Their Applications in Number Theory (Cardiff, 1995), k = 3(3ABC/2)2 that occur. Since |ABC| tends to London Math. Soc. Lecture Note Series, vol. 237, Cambridge University Press, 1997, pp. 65–85. infinity, P(k) will approach [D-G] H. DARMON and A. GRANVILLE, On the equations 2 P(ABC )=2 P(ABC) 2a in the . Since zm = F(x, y) and Axp + Byq = Czr , Bull. London a>3, we may take e = a +3 and obtain a contra- Math. Soc. 27 (1995), 513–543. diction to the Conjecture about Sums of Squares [F-G] M. FILASETA and S. KONYAGIN, On a limit point asso- and Cubes, thereby proving the proposition. ciated with the abc-conjecture, Colloq. Math. 76 (1998), 265–268. The proposition suggests that we focus on find- [G-P-Z] J. GEBEL, A. PETHÖ, and H. ZIMMER, Computing inte- ing pairs of relatively prime integers (u, v) such that gral points on Mordell’s elliptic curves, Collect. Math. P(u2 + v3) is large. Apart from the Catalan solution 48 (1997), 115–136. (u, v)=(3, 2) , a few known ones where u2 + v3 is [M] D. MASSER, Open problems, Proc. Sympos. Analytic 2 3 9 Number Theory (W. W. L. Chen, ed.), Imperial Col- plus-or-minus a perfect power are 13 +7 =2 , lege, London, 1985. 2 3 7 71 +( 17) =2 , and the following other larger [O] J. OESTERLÉ, Nouvelles approches du “théorème” de ones recently found by Beukers and Zagier: Fermat, Astérisque 161/162 (1988), 165–186. [R] P. RIBENBOIM, Catalan’s Conjecture, Academic Press, 210639282 +(76271)3 =177, 1994. [T] R. TIJDEMAN, On the equation of Catalan, Acta Arith. 2 3 7 2213459 + 1414 =65 , 29 (1976), 197–209. 153122832 + 92623 = 1137, 300429072 +(96222)3 =438, 15490342 +(15613)3 = 338. Noam Elkies communicated to me the following so- lution of the Mordell Equation:

230532 5053 =3 227;

here P(3 227)=11.05817 .... Can one find rela- tively prime u and v such that P(u2 + v3) 12? It would be good to gain enough insight to be able to offer a plausible prediction of a specific upper bound B for P of integers of the form u2 + v3, other than the Catalan solution, with u and v rel- atively prime.

202 NOTICES OF THE AMS VOLUME 47, NUMBER 2