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,
(6)
with
(7)
for ,
improving a result of Stewart and Tijdeman (1986).
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Forms, Elliptic Curves and the -Conjecture." http://www.math.columbia.edu/~goldfeld/ABC-Conjecture.pdf.Guy,
R. K. Unsolved
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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." Astérisque161/162,
165-186, 1988.Peterson, I. "MathTrek: The Amazing ABC Conjecture."
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J. "Wieferich's Criterion and the abc Conjecture." J. Number Th.30,
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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.
, there exists a constant 
such that for any three relatively
prime integers 
, 
,
satisfying
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).
as
runs over all prime divisors of 
, 
,
and 
.
Then the abc conjecture states that for all 
, there exists a constant 
such that for all 
,
or rational integers
with large height compared to the radical,
![h(p)>=r(P)+4K_l(sqrt(h(P)))/(ln[h(P)]),](/images/equations/abcConjecture/NumberedEquation4.svg)
,
improving a result of Stewart and Tijdeman (1986).
-Conjecture." 
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."