abc Conjecture
The abc conjecture is a conjecture due to Oesterlé and Masser in 1985. It states that, for any infinitesimal 
, there exists a constant 
such that for any three relatively
prime integers 
, 
, 
satisfying
 |
(1)
|
the inequality
 |
(2)
|
holds, where 
indicates that the product
is over primes 
which divide
the product 
. If this conjecture were true, it would imply Fermat's
last theorem for sufficiently large powers (Goldfeld
1996). This is related to the fact that the abc conjecture implies that there are
at least 
non-Wieferich
primes 
for some constant 
(Silverman 1988,
Vardi 1991).
The conjecture can also be stated by defining the height and radical of the sum 
as
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
runs over all prime divisors of 
, 
, and 
. Then the abc conjecture
states that for all 
, there exists a constant
such that for all 
,
 |
(5)
|
(van Frankenhuysen 2000). van Frankenhuysen (2000) has shown that there exists an infinite sequence of sums 
or rational
integers with large height compared to the radical,
![h(p)>=r(P)+4K_l(sqrt(h(P)))/(ln[h(P)]),](/National_Library/im_/https://mathworld.wolfram.com/images/equations/abcConjecture/NumberedEquation4.gif) |
(6)
|
with
 |
(7)
|
for 
, improving a result of Stewart
and Tijdeman (1986).
Cox, D. A. "Introduction to Fermat's Last Theorem." Amer. Math. Monthly 101, 3-14, 1994.
Elkies, N. D. "ABC Implies Mordell." Internat. Math. Res. Not. 7, 99-109, 1991.
Goldfeld, D. "Beyond the Last Theorem." The Sciences 36, 34-40, March/April 1996.
Goldfeld, D. "Beyond the Last Theorem." Math. Horizons, 26-31 and 24, Sept. 1996.
Goldfeld, D. "Modular Forms, Elliptic Curves and the 
-Conjecture."
http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf.
Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 75-76, 1994.
Lang, S. "Old and New Conjectures in Diophantine Inequalities." Bull. Amer. Math. Soc. 23, 37-75, 1990.
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and Discriminants."
Proc. Amer. Math. Soc. 130, 3141-3150, 2002.
Mauldin, R. D. "A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem." Not. Amer. Math. Soc. 44, 1436-1437, 1997.
Nitaq, A. "The abc Conjecture Home Page." http://www.math.unicaen.fr/~nitaj/abc.html.
Oesterlé, J. "Nouvelles approches du 'théorème' de Fermat." Astérisque 161/162, 165-186, 1988.
Peterson, I. "MathTrek: The Amazing ABC Conjecture." Dec. 8, 1997. http://www.maa.org/mathland/mathtrek_12_8.html.
Silverman, J. "Wieferich's Criterion and the abc Conjecture." J. Number Th. 30, 226-237, 1988.
Stewart, C. L. and Tijdeman, R. "On the Oesterlé-Masser Conjecture." Mh. Math. 102, 251-257, 1986.
Stewart, C. L. and Yu, K. "On the ABC Conjecture." Math. Ann. 291, 225-230, 1991.
van Frankenhuysen, M. "The ABC Conjecture Implies Roth's Theorem and Mordell's Conjecture." Mat. Contemp. 16, 45-72, 1999.
van Frankenhuysen, M. "A Lower Bound in the abc Conjecture." J. Number Th. 82, 91-95, 2000.
Vardi, I. Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, p. 66, 1991.
Vojta, P. Diophantine Approximations and Value Distribution Theory. Berlin: Springer-Verlag, p. 84, 1987.
Weisstein, Eric W. "abc Conjecture." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/abcConjecture.html
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abc conjecture