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					A Generalization of
Fermat’s Last Theorem:
The Beal Conjecture and
Prize Problem
R. Daniel Mauldin




            A
                       ndrew Beal is a Dallas banker who                   The prize. Andrew Beal is very generously of-
                       has a general interest in mathemat-              fering a prize of $5,000 for the solution of this
                       ics and its status within our culture.           problem. The value of the prize will increase by
                       He also has a personal interest in the           $5,000 per year up to $50,000 until it is solved. The
                       discipline. In fact, he has formulated           prize committee consists of Charles Fefferman,
         a conjecture in number theory on which he has                  Ron Graham, and R. Daniel Mauldin, who will act
         been working for several years. It is remarkable that          as the chair of the committee. All proposed solu-
         occasionally someone working in isolation and                  tions and inquiries about the prize should be sent
         with no connections to the mathematical world for-
                                                                        to Mauldin.
         mulates a problem so close to current research ac-
                                                                           The abc conjecture. During the 1980s a con-
         tivity.
                                                                        jectured diophantine inequality, the “abc conjec-
         The Beal Conjecture                                            ture”, with many applications was formulated by
                                                                        Masser, Oesterle, and Szpiro. A survey of this idea
         Let A, B, C, x, y , and z be positive integers with            has been given by Lang [5] and an elementary dis-
         x, y, z > 2. If Ax + B y = C z , then A, B, and C have         cussion by Goldfeld [4]. This inequality can be
         a common factor.                                               stated in very simple terms, and it can be applied
         Or, slightly restated:                                         to Beal’s problem. To state the abc conjecture, let
            The equation Ax + B y = C z has no solution in              us say that if a, b, and c are positive integers, then
         positive integers A, B, C, x, y , and z with x, y, and         N(a,b,c) denotes the square free part of the prod-
         z at least 3 and A, B, and C coprime.                          uct abc . In other words, N(a,b,c) is the product of
            It turns out that very similar conjectures have             the prime divisors of a, b, and c with each divisor
         been made over the years. In fact, Brun in his 1914            counted only once. The abc conjecture can be for-
         paper states several similar problems [1]. How-                mulated as follows:
         ever, it is very timely that this problem be raised
         now, since Fermat’s Last Theorem has just recently             For each > 0 , there is a constant µ > 1 such that
         been proved (or re-proved) by Wiles [6]. Some of               if a and b are relatively prime (or coprime) and c
         the significant advances made on some problems                 = a+b, then
         closely related to the prize problem by Darmon and
         Granville [2] are indicated below. Darmon and                           max(|a|, |b|, |c|) ≤ µN(a, b, c)1+ .
         Granville in their article also discuss some related
                                                                            Now let us show that if the abc conjecture holds,
         conjectures along this line and provide many rel-
                                                                        then there are no solutions to the prize problem
         evant references.
                                                                        when the exponents are large enough.
                                                                            Let k = log µ/log 2 + (3 + 3 ) . Let min(x, y, z) >
         R. Daniel Mauldin is Regents Professor of mathematics at
         the University of North Texas, Denton, TX. His e-mail ad-      k . Assume A, B, and C are positive integers with
         dresses are mauldin@unt.edu and mauldin@                       A and B relatively prime and such that
         dynamics.math.unt.edu.                                         Ax + B y = C z . Setting a = Ax and b = B y , we have


1436                                           NOTICES   OF THE   AMS                                 VOLUME 44, NUMBER 11
c = a + b = C z . From the abc conjecture and the             657 , 92623 + 153122832 = 1137 , 438 + 962223 =
fact that N(Ax , B y , C z ) ≤ ABC , we have                  300429072 , 338 + 15490342 = 156133 . The last
            max(Ax , B y , C z ) ≤ µ(ABC)1+ .                 five big solutions were found by Beukers and Zagier.
                                                                 Recently Darmon and Merel have shown that
If max(A, B, C) = A , then we would have                      there are no coprime solutions with exponents
                      Ax ≤ µA3+3                              (x, x, 3) with x ≥ 3 [3].
                                                                 Acknowledgment. Since I am not an expert in
or                 log µ                                      this field, I would like to thank Andrew Granville
                x≤       + 3 + 3 ≤ k,
                   log A                                      and Richard Guy for their expert help in prepar-
                                                              ing this note.
which is not the case. A similar argument for the
other two possibilities for the maximum shows that            References
our original assumption is impossible.                         [1] V. Brun, Über hypothesenbildung, Arc. Math. Naturv-
   Next let us give an explicit version of the abc con-            idenskab 34 (1914), 1–14.
jecture: If a and b are coprime positive integers and          [2] H. Darmon and A. Granville, On the equations
c = a+b, then c ≤ (N(a, b, c))2. Let us see what this              z m = F(x, y) and Axp + By q = cZ r , Bull. London
                                                                   Math. Soc. 27 (1995), 513–543.
implies for the prize problem. Suppose                         [3] H. Darmon and L. Merel, Winding quotients and
Ax + B y = C z , with x ≤ y ≤ z. Again, since Ax and               some variants of Fermat’s Last Theorem, preprint.
B y are coprime,                                               [4] D. Goldfeld, Beyond the Last Theorem, Math Hori-
     C z ≤ (N(Ax B y C z ))2 ≤ (ABC)2 < C 2(z/x+z/y+1) .           zons (September 1996), 26–31, 34.
                                                               [5] S. Lang, Old and new conjectured diophantine in-
So 1/2 < 1/x + 1/y + 1/z . Since x, y , and z are                  equalities, Bull. Amer. Math. Soc. 23 (1990), 37–75.
greater than 2, we have the following possibilities            [6] A. Wiles, Modular elliptic curves and Fermat’s Last
                                                                   Theorem, Ann. Math. 141 (1995), 443–551.
for (x,y,z): (3, 3, z > 3), (3, 4, z ≥ 4), (3, 5, z ≥ 5),
(3, 6, z ≥ 7), (4, 4, z ≥ 5) , and a finite list of other
cases.
   There are only finitely many possible solu-
tions. In 1995 Darmon and Granville [2] showed
that if the positive integers x, y, and z are such that         Andrew Beal is a number theory enthusiast re-
1/x + 1/y + 1/z < 1 , then there are only finitely              siding in Dallas, Texas. He grew up in Lansing,
many triples of coprime integers A, B, C satisfying             Michigan, and attended Michigan State Uni-
Ax + B y = C z . Since each of x, y, and z is greater           versity. He has a particular interest in some of
than 2, then 1/x + 1/y + 1/z < 1 unless                         Fermat’s work and has spent many, many
x = y = z = 3. But Euler and possibly Fermat knew               hours thinking about Fermat’s Last Theorem.
there are no solutions in this case. So for each                He believes that Fermat did possess a relatively
triple x, y, and z , all greater than 2, there can be           simple non-geometry-based proof for FLT, and
                                                                he continues to search for it. He also believes
only finitely many solutions to the diophantine
                                                                that Fermat had a method of solution for Pell’s
equation Ax + B y = C z .
                                                                equation that remains unknown and that was
   Related problems. What happens if it is only re-
                                                                a function of the squares whose sum equals the
quired that x, y, and z be ≥ 2 and at least one of
                                                                coefficient.
them is greater than 2 and A, B, and C are coprime?
                                                                   Andrew is forty-four years old. He and his
There is a detailed analysis in [2] of those cases
                                                                wife, Simona, have five children. He is the
where x, y, z ≥ 2 and 1/x + 1/y + 1/z > 1 .
                                                                founder/chairman/owner of Beal Bank, Dal-
   What happens if we require only that
                                                                las’s largest locally owned bank. He is also the
1/x + 1/y + 1/z < 1 and A, B, and C are coprime?
                                                                recent founder/CEO/owner of Beal Aerospace,
This problem is also discussed by Darmon and
                                                                which is designing and building a next-gener-
Granville. In fact, they have formulated
                                                                ation rocket for launching satellites into earth
The Fermat-Catalan Conjecture. There are only                   orbits.
finitely many triples of coprime integer powers                    Beal Bank, Toyota, and the Dallas Morning
xp , y q , z r for which                                        News are the primary sponsors of the Dallas
                                1 1 1                           Regional Science and Engineering Fair. Beal
             xp + y q = z r with + + < 1.
                                p q r                           Bank is also a primary sponsor of the Dallas
                                                                Area Odyssey of the Mind Competition. An-
So far, as mentioned in [2], ten solutions have been            drew Beal has been a major benefactor for the
found. The first five are small solutions. They are             mathematics program at the University of
1 + 23 = 32 , 25 + 72 = 34 , 73 + 132 = 29 , 27 + 173 =         North Texas through his substantial scholar-
712 , 35 + 114 = 1222 .                                         ships for graduate students and for students in
   Also five large solutions have been found:                   the Texas Academy of Mathematics and Science.
177 + 762713 = 210639282 , 14143 + 22134592 =


DECEMBER 1997                                               NOTICES   OF THE   AMS                                        1437

				
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