THE ABC CONJECTURE HOME
PAGE
La conjecture abc
est aussi difficile que la conjecture
... xyz. (P. Ribenboim) (read
the story)
The abc
conjecture is the most important unsolved
problem in diophantine analysis. (D. Goldfeld)
Created and maintained by Abderrahmane
Nitaj
Last updated March 21, 2013
Proof of the abc
Conjecture?
On August 30, 2012, Shinichi Mochizuki, a
mathematician at Kyoto University in Japan, published four papers on
the Internet claiming to prove the abc conjecture.
Shinichi
Mochizuki home page
For a natural number, let rad(n)
be the product of all distinct prime divisors of n.
E.g. if n=2^{5}
× 3^{7} × 11 × 17^{2}
then rad(n)=2
× 3 × 11 × 17=1122.
Given any >
0, there exists a constant C_{}>
0 such that for every triple of positive integers a,b, c, satisfying
a+b=c
and gcd(a,b)=1 we have
c C_{}
(rad(abc))^{1+}.
The abc conjecture was first formulated by
Joseph
Oesterlé [Oe] and David
Masser [Mas]
in 1985. Although the abc conjecture seems
completely out of reach,
there are some results towards the truth of this conjecture.
 1986, C.L. Stewart and R. Tijdeman [SteTi]: c
< exp{ C_{1}rad(abc)^{15}
},
 1991, C.L. Stewart and Kunrui Yu [SteYu1]: c
< exp{ C_{2}rad(abc)^{2/3+}
},
where C_{1} is an absolute constant, C_{2}
and C_{3}
are positive effectivley computable constants in terms of .  2007,
K. Gyory new
results on the abc conjecture:
To Index
 The abc
theorem for polynomials. For
a polynomial P with complex coefficients let N_{0}=N_{0}(P)
be the number of distinct roots of P. A theorem of
Stothers [Sto]
and Mason [Ma] states that if A, B, C are relatively prime
polynomials
such that A+B=C, then
max(deg(A), deg(B), deg(C)) N_{0}(ABC)1.
This is the well known abc theorem for polynomials.
On the other
hand, we have (see [Va])
min(deg(A), deg(B), deg(C)) N_{0}(ABC)2.
 The abc
conjecture for binary forms.
It is shown in [Lan2] that the abc conjecture
implies the following
conjecture.
Let F(X,Y) be a homogenous polynomial with integer
coefficients
and no repeated linear factors. For any
> 0, there exists a constant C_{,F}
such that for any coprime integers m and n,
max(m,n)
C_{,F}
(rad(mnF(m,n)))^{deg(F)+}.
Conversely, this conjecture implies the abc
conjecture when F(X,Y)=X+Y.  The nterm abc
conjecture for integers.
In 1994, Browkin and Brzezinski [BrBrz] proposed the following
conjecture.
Given any integer n > 2 and any >
0, there exists a constant C_{n,}
such that for all integers a_{1}, ..., a_{n}
with a_{1}+...+
a_{n}=0, gcd( a_{1},..., a_{n})=1
and no proper
zero subsum, we have
max(a_{1},...,a_{n})
C_{n,}(rad(a_{1}
× ... × a_{n}))^{2n5+}.
 Baker's abc
conjecture for integers.
In 1996, Alan Baker [Ba] proposed the following version of the abc
conjecture in connection with the theory of linear forms in logarithms.
Given any >
0, there exists a constant C_{}>
0 such that for every triple of positive integers a,b, c, satisfying
a+b=c
and gcd(a,b)=1 we have
c C_{}(^{}rad(abc))^{1+},
where denotes the
number of distinct prime factors of abc.  The abc
conjecture for number fields.
Let K be an algebraic number
field and let V_{K}
denote the set of primes on K,
that is, any v in V_{K}
is an equivalence class of nontrivial norms on K
(finite
or infinite). Let x_{v}=N_{K/Q}(P)^{vP(x)}
if v is a prime definied by a prime ideal P
of the ring of
integers O_{K}
in K and v_{P}
is the corresponding valuation, where N_{K/Q}
is
the absolute norm. Let x_{v}=g(x)^{e}
for
all nonconjugate embeddings g: K
> C with e=1 if g
is real and e=2 if g is
complex. Define the height of any
triple a,b,c in K^{*}
to be
H_{K}(a,b,c)
= _{v
in VK} max(a_{v},
b_{v}, c_{v}),
and the radical (or conductor) of (a,b,c) by
rad_{K}(a,b,c)
= _{
P in IK(a,b,c)}N_{K/Q}(P),
where I_{K}(a,b,c)
is the set of all prime ideals P
of O_{K} for
which a_{v}, b_{v}
, c_{v} are not equal. Let D_{K/Q}
denote the discriminant of K.
 The abc
conjecture for algebraic number fields.For
any > 0, there
exists a positive constant C_{K,}
such that for all a,b,c in K^{*}
satisfying a+b+c=0, we
have
H_{K}(a,b,c)
< C_{K,}(rad_{K}(a,b,c))^{1+}.
 The uniform abc
conjecture.For any >
0, there exists a positive constant C_{}
such that for all a,b,c in K^{*}
satisfying a+b+c=0, we
have
H_{K}(a,b,c)
< C_{}^{[K:Q]}(D_{K/Q}rad_{K}(a,b,c))^{1+}.
K.
Gyory new results on the uniform abc conjecture
for number fields:  The abc
theorem for nonarchimedean meromorphic
function fields. Let K
be a nonarchimedean algebraically closed field of characteristic zero.
Let a(z), b(z), c(z) be entire functions in K
without
common zeros and not all constants satisfying a+b=c.
In 2000, Hu
and Yang [HuYa] showed that
max{T(r,a), T(r,b), T(r,c)} <
N(r,1/(abc))log(r)+O(1),
where T and N are functions
related to Nevanlinna's value
distribution theory (see [HuYa] and [HuYa3]). StothersMason's abc
theorem for polynomials is an application of this result.
 The kterm abc
theorem for nonarchimedean
meromorphic function fields. Let K
be a nonarchimedean algebraically closed field of characteristic zero.
Let f_{j}(z), j=0...k, be k
entire functions in K
without common zeros, not all constants and no proper subsum is equal
to
0 satisfying f_{0}+f_{1}+....+f_{k}
= 0.
In 2002, Hu and Yang [HuYa3] showed that
max{T(r,f_{j})} <
N(r,1/f_{0}, 1/f_{1},...,
1/f_{k}))k(k1)log(r)/2+O(1),
where T and N are functions
related to Nevanlinna's value
distribution theory (see [HuYa3]). StothersMason's abc
theorem
for polynomials is an application of this result with k=2.
 HuYang's kterm abc
conjecture for integers.
Let a be a nonzero integer with the factorization a=p_{1}^{i1}...p_{n}^{in}
where p_{1},...,p_{n}
are distinct primes. Define
the kradical of a to be
r_{k}(a)=_{pja}
p_{j}^{min(ij,k)}.
In 2002, Hu and Yang [HuYa3] proposed the following conjecture.
Let a_{i}, i=0...k, be nonzero
integers without common factor
and no proper subsum is equal to 0 such that
a_{0}+.....+a_{k}
=0.
Then for >0,
there exists a constant C(k,)
such that
maxa_{i} < C(k,)R(a_{0}...a_{k})^{1+},
where
R(a_{0}...a_{k})
= _{i}
r_{k1}(a_{i}).
If k=2, this corresponds
to the abc
conjecture.
To Index
Pierre de
Fermat
 Fermat's Last Theorem.
Fermat's conjecture,
known as Fermat's Last Theorem states that The equation x^{n}+y^{n}=z^{n}
has no non trivial integer solution for n > 2 and has been
proved by A.
Wiles. The abc conjecture implies the asymptotic form of the Fermat
Last
Theorem, i.e. that there are only finitely many solutions to the
equation
x^{n}+y^{n}=z^{n}
with gcd(x,y,z)=1 and n> 3.
Andrew
Wiles
 The generalized Fermat
equation.
The abc
conjecture implies [Ni4, Ti] that for given positive integers A, B, C,
the generalized Fermat equation Ax^{r}+By^{s}=Cz^{t}
has only finitely many solutions in integers x, y, z, r, s, t
satisfying
gcd(x,y,z)=1
and 1/r+1/s+1/t< 1 . Note that Darmon and Granville [DarGr]
proved
that if r, s, t are fixed with 1/r+1/s+1/t< 1 then the equation
Ax^{r}+By^{s}=Cz^{t}
has only finitely many solutions in pairwise coprime integers x,y,z.
 Wieferich
primes. A prime p is called a Wieferich prime if p^{2}
divides
2^{ p1}1. Such primes are related to the first
case of Fermat's
last theorem. 1093 and 3511 are the only known Wieferich primes below
4,000,000,000,000.
J. H. Silverman [Si] proved that the abc conjecture implies the
following
open problem.
Given a positive integer a> 1. There exist infinitely
many primes
p such that p^{2} does not divide a^{p1}1.
 The ErdösWoods
conjecture. It was
conjectured by Erdös and Woods that there exists an absolute
constant
k > 2 such that for every positive integers x and y, if
rad(x+i)=rad(y+i)
for i=1,2,...,k then x=y. No examples with different x and y are known.
Langevin [Lan1, Lan2] proved that the abc conjecture implies the
ErdösWoods
conjecture with k=3 except perhaps a finite number of counter examples.
 Arithmetic
progressions with the
same prime divisors.
For each quadruple (x,y,d,d') of positive integers where (x,d) and
(y,d')
are different, satisfying gcd(x,d)=gcd(y,d')=1, let K=K(x,y,d,d') be
the
largest positive integer K for which
rad(x+id)=rad(y+id') for i=0,...,K1.
This somewhere extends the ErdösWoods conjecture to
arithmetic
progressions.
It is shown in [BalLanShoWal] that the correctness of the abc
conjecture
implies that for each pair (d,d') of positive integers, the set of
pairs
(x,y)
satisfying K(x,y,d,d') > 2 is finite, and the set of quadruples
(x,y,d,d')
satisfying K(x,y,d,d') > 4 is also finite.
 Hall's conjecture.
The abc conjecture implies
the following weak form of the Hall conjecture [Ni4, Sch]:
Given any >
0, there exists a constant C>
0 such that for every positive integers x and y such that x^{3}
 y^{2} is nonzero then x^{3}
 y^{2}> C
max(x^{3}, y^{2})^{1/6}.  Erdös'
conjecture on consecutive
powerful numbers. A positive integer n is powerful if for
every
prime p dividing n, p^{2} also divides n. Every
powerful number
can be written as a^{2}b^{3}
where a and b are positive
integers. Erdös' conjecture asserts
that there do
not exist three consecutive powerful integers. The
abc conjecture implies
the weaker assertion that the set of triples of consecutive powerful
integers
is finite.
 Brown
Numbers and Brocard's Problem. Pairs of numbers (m,n)
satisfying Brocard's
Problem n!+1=m^{2} are called Brown pairs. The abc
conjecture implies
that there are only finitely many such pairs [Ov]. This has been
generalized
to the number of integer solutions of the equations
(x!)^{n}+1=y^{m}
[Ni4] and x!+B^{2}=y^{2} for
general B [Da].
 Szpiro's conjecture
for elliptic
curves.
This conjecture states that the minimal discriminant of an elliptic
curve
is controlled by its conductor, namely,
Given any >
0, there exists a constant C_{}>
0 such that for every elliptic curve with minimal discriminant
and conductor N, we have 
< C_{}N^{6+}.
It has been proved (e.g. in [Oe] and [Ni4]) that this conjecture
follows from the abc conjecture.
The abc conjecture has other consequences on the arithmetic of elliptic
curves, via the very important family of FreyHellegouarch
curves.
 Mordell's conjecture.
This 1922 conjecture
asserts that any curve of genus larger than 1 defined over a number
field K has only finitely many rational points in K.
This conjecture
is now a theorem after the work of G. Faltings (1984). In [EL] it is
shown
that the truth of the abc conjecture for number fields implies the
truth
of the Mordell conjecture over an arbitrary number field. Morever, if
one
could prove the abc conjecture with an explicit constant C,
then one would have explicit bounds on the heights of the rational
points
in Mordell's conjecture (see also [Fr3]).
 Squarefree values of
polynomials.
It has
not been shown that there exists an irreducible polynomial F(X) with
rational
coefficients in one variable of degree at least 5 such that F(n) is
squarefree
for infinitely many integers n. Browkin, Filaseta, Greaves and Schinzel
[BrFiGrSc] proved that the abc conjecture gives a positive answer
for cyclotomic
polynomials _{n}(X)
and for (X^{n}1)/(X1).
 Roth's theorem.
In
1955, Klaus Roth proved
that for every algebraic number
, the diophantine equation p/q
< 1/q^{2+},
with > 0, has
only finitely many solutions. Applying the abc conjecture, E. Bombieri
[Bo] proved in 1994 a stronger effective version of this theorem,
namely
that one has the inequality p/q
> 1/q^{2+k} for all but a finite number of
fractions p/q in lowest
form, where k= C(log q)^{1/2}(log(log q))^{1}
for some
constant C depending only on .
(See also [Fr3]).
 Dressler's conjecture.
The conjecture of
Dressler states that between any two different positive integers having
the same prime factors there is a prime. Cochrane and Dressler [CoDr]
proved that the abc conjecture implies that for any >
0 there is a constant C
such that if a< b are positive integers having the same prime
factors,
then
ba> C a^{1/2}.
 Siegel zeros.
Let
L(s,)
be the Dirichlet
Lfunction of
characters of the form (d/.) where d < 0 is a fundamental
discriminant
of an imaginary quadratic field. Real solutions of the equation
L(s,(d/.))=0
in the interval 1c/(log d) < s < 1 for a small constant
c > 0 are
called Siegel zeros. Let h(d) be the class number of Q((d)^{1/2}).
Granville and Stark [GrSt] proved that the uniform abc conjecture for
number fields implies that
h(d) > ( /3+o(1))d^{1/2}(log
d)^{1}(1/a),
where the sum runs over all
reduced quadratic
forms ax^{2}+bxy+cy^{2} of
discriminant d. It is known
since Mahler that if this holds, then the Dirichlet Lfunction
L(s,(d/.))
has no Siegel zero. Consequently, "ABC implies no Siegel Zero".
 Power freevalues of
polynomials.
Langevin
noted in [Lan2] the following conjecture which is a consequence of the
abc
conjecture.
Let F(X) be a polynomial with integer coefficients and no
repeated
roots. For any
> 0, there exists a constant C_{,F}
such that for any integer n,
n^{deg(F)1}
C_{,F} rad(F(n)).
 Counting
squarefreevalues of
polynomials.
Let f(x,y) be a homogenous polynomial with integer coefficients and no
repeated linear factors. Let B be the largest integer which divides
f(m,n)
for all pairs of integers m,n and set B' = B/rad(B). Granville [Gr3]
proved
that the abc conjecture implies that there are O(c_{f}MN)
pairs
of positive integers m < M, n < N for which f(m,n)/B' is
squarefree
as M, N tend to infinity, where c_{f} is a constant
which depends
only on f. A similar result exists for a polynomial f(x) with integer
coefficients
and no repeated roots.
 Bounds for the order
of the
TateShafarevich group.
Let E be an elliptic curve over Q with
TateShafarevich goup
and conductor
N. The conjecture of Goldfeld and Zspiro asserts that
for
every > 0, there
exists a positive constant C
such that 
N^{1/2 + }.
Assuming the Birch
and SwinnertonDyer conjecture, it is shown in [GoSz] that this
conjecture
is equivalent to the Szpiro conjecture for modular elliptic curves.
 Vojta's height
conjecture for
curves. Vojta
formulated in [Vo2] a geneneral conjecture on algebraic points of
bounded
degree on a smooth complete variety X over a global field of
characteristic
zero. He then showed that this conjecture implies the abc conjecture.
Conversely,
van Frankenhuysen [Fr4] proved that the abc conjecture implies Vojta's
height conjecture for curves, i.e. when X is onedimensional.
 The powerful part of
terms in
binary recurrence
sequences. Every positive integer a can be written in the
form a=sq,
where s is squarefree, q is powerful and gcd(s,q)=1. The part q=w(a) is
called the powerful part of a. Let P > 0, Q be integers such
that D=P^{2}4Q
> 0 and
gcd(P,Q)=1. Let U_{0}=0, U_{1}=1
and V_{0}=2,
V_{1}=P. For each n > 1, let U_{n}=PU_{n1}QU_{n2}
and V_{n}=PV_{n1}QV_{n2}.
Ribenboim and Walsh
[RiWa] proved that the abc conjecture implies that the sets
{n > 0, w(U_{n})
> U_{n}} and
{n > 0, w(V_{n}) > V_{n}}
are finite. It follows that U_{n} and V_{n}
are powerful
for only finitely many terms.
 Greenberg's conjecture.
Let p be a prime
number and K a totally real number field. Denote _{p}(K)
and _{p}(K) the
Iwasawa invariants of the cyclotomic Z_{p}extension
of
K. The pseudonull conjecture of Greenberg (1976) asserts that
these two
invariants vanish for all p and all K. Assuming the truth of the
abc conjecture
for quadratic number fields for which the norm of a fundamental unit is
1, Ichimura [Ic] proved that there exist infinitely many primes p such
that _{p}(K)=0.
 Exponents of class
groups of
quadratic fields.
Given positive numbers g and x. Let N(x) be the number of quadratic
number
fields Q(d^{1/2}) with
0< d< x whose order of the
class group is divisible by g. It has been proved that N(x) is
infinite,
but no quantitative bound is known. R. Murty [Mur] showed that the abc
conjecture enables us to count N(x), namely that for any
> 0, there exists a positive constant C
such that N(x) > C
x^{k+} where
k=1/g if d < 0 and k=1/2g if d > 0.
 Limit points.
Let
a,b,c be positive integers
satisfying a+b=c and gcd(a,b)=1. Define L(a,b) by
L(a,b)=log(c)/log(rad(abc)).
It is shown in [GreNi] that the limit points of the sequence (L(a,b))
fill the interval [1/3, 36/37]. On the other hand, it is shown in
[FiKo]
that there exists a limit point with 1
L<
3/2. Note that the abc conjecture can be rephrased to state that the
sequence
(L(a,b)) is bounded with greatest limit point 1.
 Fundamental units of
certain
quadratic and biquadratic
fields: added to this page on June 23, 2000. For a
positive integer
M
let N_{1} = (M+(M^{2}+/4)^{1/2})/2,
and
N'_{1}
= (M(M^{2}+/4)^{1/2})/2. For
any positive integer
n,
put
g_{n} = N_{1}^{n}+N'_{1}^{n},
h_{n}
= (N_{1}^{n}N'_{1}^{n})/(M^{2}+/4)^{1/2},
N_{2}=h_{2n+1}+(h^{2}_{2n+1}1)^{1/2}
and N_{3}=g_{2n+1}/M+(g^{2}_{2n+1}/M^{2}1)^{1/2}.
Katayama [Ka] showed that if the abc conjecture is valid, then N_{2}
is the fundamental unit of the real quadratic field Q((h^{2}_{2n+1}1)^{1/2}),
N_{3}
is the fundamental unit of the real quadratic field
Q((g^{2}_{2n+1}/M^{2}1)^{1/2}),
and {N_{1}, N_{2}, N_{3}}
is a fundamental system
of units of the real bicyclic biquadratic field Q((M^{2}+/4)^{1/2},
(h^{2}_{2n+1}1)^{1/2})
except for finitely many
integers n.
 The SchinzelTijdeman
conjecture
: added to this
page on April 13, 2001. This conjecture asserts that if a
polynomial
P(x) with rational coefficients has at least three simple zeros, then
the
Diophantine equation P(x)=y^{2}z^{3}
has only finitely
many nontrivial solutions in integers x, y, z. Walsh [Wa2] proved that
the abc conjecture implies this conjecture.
 Lang's conjecture
(1978): added
to this page on
April 25, 2002. Let K be a number field. Then there
exists a constant
C(K)>0 such that if E/K is an elliptic curve and P is
nontorsion point
of the MordellWeil group E(K), then H(P)>C(K)log N_{K/Q}D_{E/K}
where D_{E/K} is the minimal discriminant of E/K,
H(P) is the canonical
height of P and N_{K/Q} is the norm. This
conjecture of Lang follows
from the ABC conjecture (see [HSi]). To be more precise, Hindry and
Silverman
proved that H(P)>(20sz )^{8[K:Q]}10^{1.14sz}log
N_{K/Q}D_{E/K}
where sz is the Szpiro ratio
logN_{K/Q}D_{E/K}
sz = ^{____________________}.
logN_{K/Q}F_{E/K}
and F_{E/K} is the conductor of E/K.
 Lang's Integral Point
Conjecture
(1978) : added
to this page on February 19, 2003 (pointed out by J.H. Silverman). Let
K be a number field and let S be a set of primes of K. Then there exist
constants C1 and C2(K) so that if E/K is an elliptic curve given by a
(relatively)
minimal Weierstrass equation, then the number of Sintegral points in
E(K)
is bounded by C1× C2^{(1+#S+rank E/K)}.
Silverman
[Si2] proved
that the integral point conjecture of Lang is a consequence of Lang's
height
lower bound conjecture (25). So the integral point conjecture is also a
consequence of the ABC conjecture.
 Rounding reciprocal
square
roots: added on May
23, 2005. Let
= x^{(1/2)} where x is a positive real number. To
get the correctly
rounded in a floating
point system with p signifcant bits, one may have to compute the 3p+1
leading
bits of x^{(1/2)}. In 2004, Croot, Li and Zhu
[CrLiZh] showed
that, assuming the abc conjecture, the number of the leading bits could
be reduced to 2p.
 The abc(k,m)
conjecture for integers : added on May
23, 2005.
Let k >1 and n >0
be integers with the factorization n=p_{1}^{i1}...p_{n}^{in}
where p_{1},...,p_{n}
are distinct primes. Define
the k'th radical of n to be
n_{k}(n)=_{pjijn}
p_{j}^{UB(ij/k)},
where UB(x) is the smallest integer greater than or
equal to x.
In 2002, Broughan [Broug] proposed the following conjecture.
Let a, b, c be positive integers without common factor
such that a+b=c.
There exists a positive constant C(k,m) such that
c < C(k,m)n_{k}(abc)^{1+1/m}.
This conjecture is a wekened form of the abc conjecture.
 The diophantine
equation p^{v}p^{w}=q^{x}q^{y}
: added on May
23, 2005.
In 2003, Luca [Lu] showed that assuming the abc
conjecture, the diophantine equation p^{v}p^{w}=q^{x}q^{y}
has only finitely many positive integer solutions p, q,
v,w,x,y where p and q
are distinct prime numbers.
 The number of
quadratic fields
generated by a polynomial : added on May
23, 2005.
In 2003, Cutter, Granville an Tucker [CuGrTu] showed that the
abc conjecture implies the following conjecture.
If a polynomial f(x) with integer coefficients has
degree larger than 1 and no repeated roots, then there are
approximately N distinct quadratic fields amongst Q(f(j)^{1/2})
for j=1, ... N.
 The ideal
Waring’s Theorem : added on March
21, 2013.
For a positive integer k>1, let g(k) be the smallest positive
integer g such that any integer is
the sum of g elements of the form x^{k} with x
>0. For example, according to Lagrange's Theorem g(2)=4 and
according to Wieferich's Theorem g(3)=9 (see [Wal1, Wal2]).
The ideal Waring’s Theorem is the 1853 conjecture that
asserts that
For any k>1, g(k)=2^{k}+[(3/2)^{k}]2.
In a personnal comunication to M. Waldschmidt (see [Wal1, Wal2]), S.
David proved that the ideal Waring’s Theorem is a consequence
of the abc conjecture for sufficiently large k.
To Index
For any triple of positive integers a, b, c satisfying a+b=c and
gcd(a,b)=1
let
log(c)
= (a,b,c) = ^{________________________},
log(rad(abc))
and
log(abc)
= (a,b,c) = ^{_________________________}.
log(rad(abc))
Triples satisfying
> 1.4 or > 4 are respectively
called good abcexamples and good abcSzpiroexamples.
For (nonzero) algebraic numbers a, b, c such that a
+ b = c, let
K=Q(a/c) and
log(H_{K}(a,b,c))
= (a,b,c) = ^{________________________________________}.
logD_{K/Q}+log(rad_{K}(a,b,c))
Triples satisfying
> 1.5 are called good algebraic abcexamples.
Authors of good abcexamples:
J.Bo.: Johan Bosman
N.B. Niclas Broberg
J.B.J.B. : Jerzy Browkin and Juliusz Brzezinski
T.D. : Tim Dokchitser
N.E.J.K. : Noam Elkies and Joe Kanapka
G.F. : Gerhard Frey
X.G. : Xiao Gang
M.H. : Mathias Hegner
A.N. : Abderrahmane Nitaj
E.R. : Eric Reyssat
H.R.P.M. : Herman te Riele and Peter Montgomery
T.S.A.R.: Traugott Schulmeiss and Andrej Rosenheinrich
T.S. : Traugott Schulmeiss
K.V. : Kees Visser
B.W. : Benne M.M. de Weger
Table I. The top ten good abcexamples
log(c) = (a,b,c) = ^{_____________________}, log(rad(abc))

No 
a 
b 
c 

Author 
1. 
2 
3^{10} × 109 
23^{5} 
1.62991 
E.R. 
2. 
11^{2} 
3^{2} × 5^{6}
×
7^{3} 
2^{21} × 23 
1.62599 
B.W. 
3. 
19 × 1307 
7 × 29^{2} × 31^{8} 
2^{8} × 3^{22}
× 5^{4} 
1.62349 
J.BJ.B 
4. 
283 
5^{11} × 13^{2} 
2^{8} × 3^{8}
×
17^{3} 
1.58076 
J.BJ.B, A.N 
5. 
1 
2 × 3^{7} 
5^{4} × 7 
1.56789 
B.W. 
6. 
7^{3} 
3^{10} 
2^{11} × 29 
1. 54708 
B.W. 
7. 
7^{2} × 41^{2}
× 311^{3} 
11^{16} × 13^{2}
× 79 
2 × 3^{3} × 5^{23}
× 953 
1.54443 
A.N. 
8. 
5^{3} 
2^{9} × 3^{17}
× 13^{2} 
11^{5} × 17 × 31^{3}
× 137 
1.53671 
P.MH.R 
9. 
13 × 19^{6} 
2^{30} × 5 
3^{13} × 11^{2}
× 31 
1.52700 
A.N. 
10. 
3^{18} × 23 ×
2269 
17^{3} × 29 × 31^{8} 
2^{10} × 5^{2}
× 7^{15} 
1.52216 
A.N 
de
Smit's
HTML complete list of good abcexamples. 
[PDF]
[DVI] [PS] complete list of all known
good abcexamples. 
[PDF]
[DVI] [PS] T.
Dokchitser's PDF, DVI or PS list of new good abcexamples. 
Table II. The top ten good
abcSzpiroexamples
log(abc) = (a,b,c) = ^{__________________________}. log(rad(abc))

No 
a 
b 
c 

Author 
1. 
13 × 19^{6} 
2^{30} × 5 
3^{13} × 11^{2}
× 31 
4. 41901 
A.N. 
2. 
2^{5} × 11^{2}
× 19^{9 } 
5^{15} × 37^{2}
× 47 
3^{7} × 7^{11}
× 743 
4.26801 
A.N. 
3. 
2^{19} × 13 ×
103 
7^{11} 
3^{11} × 5^{3}
× 11^{2} 
4.24789 
B.W. 
4. 
19^{8} × 43^{4}
× 149^{2} 
2^{15} × 5^{23 }×
101 
3^{13} × 13× 29^{2}
× 37^{6}
× 911 
4.23181 
T.D. 
5. 
2^{35} × 7^{2}
× 17^{2} ×
19 
3^{27} × 107^{2 } 
5^{15} × 37^{2}
× 2311 
4.23069 
A.N. 
6. 
3^{18} × 23 ×
2269 
17^{3} × 29 × 31^{8} 
2^{10} × 5^{2}
× 7^{15 } 
4.22979 
A.N. 
7. 
17^{4} × 79^{3}
× 211 
2^{29} × 23 × 29^{2} 
5^{19} 
4.22960 
A.N. 
8. 
5^{14} × 19 
2^{5} × 3 × 7^{13} 
11^{7} × 37^{2}
× 353 
4.22532 
A.N. 
9. 
2^{7} × 5^{4}
×
7^{22} 
19^{4} × 37× 47^{4}
× 53^{6} 
3^{14} × 11× 13^{9}
× 191 ×
7829 
4.21019 
T.D. 
10. 
3^{21} 
7^{2} × 11^{6}
× 199 
2 × 13^{8} × 17 
4.20094 
A.N. 

[PDF]
[DVI] [PS] complete list of all
known good abcSzpiro examples. 
[PDF]
[DVI] [PS] T.
Dokchitser's PDF, DVI or PS list of new good abcSzpiro
examples. 
Table III. The top ten good purely
algebraic abcexamples
over K=Q(d)
log(H_{K}(a,b,c)) = (a,b,c) = ^{_____________________________________}. logD_{K/Q}+log(rad_{K}(a,b,c)))

No 
Equation 
a 
b 
c 
w 

Author 
1. 
w^{2}w3=0 
w 
(w+1)^{10}(w1) 
2^{9}(w+1)^{5} 
(1+13^{1/2})/2 
2.029229 
T.D. 
2. 
w_{i}^{3}2w_{i}^{2}+4w_{i}4=0,
i=1,2,3 
(w_{3}w_{2})w_{1}^{52} 
(w_{1}w_{3})w_{2}^{52} 
(w_{2}w_{1})w_{3}^{52} 
de
Weger's example 
1.920859 
B.W. 
3. 
w_{i}^{3}+3w_{i}1=0,
i=1,2,3 
(2w_{1}+1)(8w_{1}3)w_{1}^{16} 
(2w_{2}+1)(8w_{2}3)w_{2}^{16} 
(2w_{3}+1)(8w_{3}3)w_{3}^{16} 
Dokchitser's
example 
1.918150 
T.D. 
4. 
w^{3}+3w^{2}4w+1=0 
(w^{2}+4w1)^{3} 
(w^{2}+4w1)^{11}(w^{2}+3w2)^{11} 
(w^{2}+3w2)^{8} 
Dokchitser's
example 
1.834740 
T.D. 
5. 
w^{2}2=0 
w^{17} 
(1w)^{5}(3w) 
(1+w)^{5}(3+w) 
2^{1/2} 
1.768124 
N.B. 
6. 
w^{2}5w4=0 
(2w+1)^{3}(2w13)^{2} 
(10w+7)^{10}(w6)^{17}(8w7) 
(10w+7)^{7}(w+1)^{5}(2w11)^{15} 
(5+41^{1/2})/2 
1.753452 
T.D. 
7. 
w^{3}+3w+1=0 
w^{14}(w2) 
(w^{2}w+1)^{5} 
w^{2}(w^{2}+1)^{22} 
Dokchitser's
example 
1.751018 
T.D. 
8. 
w^{2}w4=0 
(2w5)^{4}(w+1)^{5} 
(2w+3)^{4}(w2)^{5} 
3^{5} 
(1+17^{1/2})/2 
1.712274 
T.D. 
9. 
w^{2}w13=0 
(w+3)^{4}(w+1)^{3} 
(w4)^{4}(w2)^{3} 
3 × 7^{6} 
(1+53^{1/2})/2 
1.719820 
T.D. 
10. 
w^{2}5w+2=0 
(2w1)^{8}(w4)^{5} 
(w1)^{5} 
3^{5}(2w1)^{4} 
(5+17^{1/2})/2 
1.719820 
T.D. 
11. 
w^{2}+w+2=0 
(12w) 
(1w)^{13} 
w^{13} 
(1+(7)^{1/2})/2 
1.707221 
N.B. 
12. 
w^{2}w1=0 
2^{4}× 3^{2}×(12w) 
w^{12} 
(1w)^{12} 
1/2+5^{1/2}/2 
1.697797 
J.Bo. 
... 
... 
... 
... 
... 
... 
... 
... 
.. 
w^{2}2=0 
1 
(1+w)^{14} 
13^{2}× (1+w)^{7}w^{3} 
2^{1/2} 
1.561437 
N.B. 
.. 
w^{2}7=0 
(83w)^{2} (52w) 
(83w)^{7} (3w)^{3}
(5+2w)^{12} 
(43w)^{4} 
7^{1/2} 
1.528940 
N.B. 
.. 
w^{2}6=0 
7^{2}(5+2w)^{9}
(2w)^{9} (3w)
(1+w) (1w) 
1 
(5+2w)^{8} 
6^{1/2} 
1.518102 
N.B. 
Broberg's
list of good algebraic abcexamples. 
Dokchitser's
list of good algebraic abcexamples. 
Table IV. New good abcexamples,
sorted by date
log(c) = (a,b,c) = ^{_____________________}, log(rad(abc))

New authors:
J.D. : Jeroen Demeyer B.S. : Bart de Smit H.W.L. : Hendrik W. Lenstra W.J.P. : Willem Jan Palenstijn F.R. : Frank Rubin I.C. : Ismael Jimenez Calvo J.W. : Jarek Wroblewski

Date 
a 
b 
c 

Author 
May 22, 2010 
2^{5}× 5^{5}×
7^{5}×11^{3}×29^{2}×347^{2} 
3^{8} ×97^{8}
×1091^{2} 
13^{12} × 19^{7}
× 2939 
1.4128 
F.R. 
April 23, 2010 
5^{4}×19^{13}×103 
2^{13} ×13^{9}
×29×2441×7673^{2} 
3^{19} ×11^{4}
×463^{5} 
1.4494 
F.R. 
March 05, 2010 
11×19^{8}×23×
67^{5}×1877 
3^{12}×47^{3}×83^{5}×113^{3}

2^{17}×5^{22}×1019^{2} 
1.4019 
F.R. 
February 25, 2010 
311^{7}× 6029 
2^{45} ×5^{6}
×839^{2} 
3^{8} ×7^{3}
×17^{6} ×43^{2}
×157^{3} 
1.4229 
F.R. 
February 12, 2010 
2^{37}× 3^{12}×2109^{3} 
5^{13} ×13^{15}
×2939 
7^{23} × 11×
793345871 
1.4121 
F.R. 
January 28, 2010 
2^{14}× 3^{6}×
42487^{3} 
5^{14} ×29^{12}
×83 
7^{8} × 11^{3}
× 47^{7} × 4610911 
1.4126 
F.R. 
May 02, 2009 
31^{2}× 61^{7}×
3889 
3^{23} ×11^{7}
× 151^{3} ×173 
2^{23} × 5^{6}
× 7^{3}× 83^{3}×
349^{3} 
1.4150 
J.W. 
April 22, 2009 
7^{5}× 61 
2^{13} ×13^{7}
× 17^{3} × 4229^{3}

3^{13} × 5^{8}
× 11^{3}× 53× 73^{2}
×89^{2}× 103 
1.4077 
J.W. 
March 01, 2009 
41^{2}× 59^{2} 
3^{8} ×7^{6}
×13^{8} ×1831 
2^{12} × 5^{4}
× 76651^{3} 
1.4072 
F.R. 
January 02, 2009 
3^{5}× 5^{15}×
13^{5} 
7^{10} ×79^{5}
× 35323 
2^{11} × 73^{7}
× 83^{2}× 197 
1.4091 
F.R. 
December 25, 2008 
2^{12}× 13^{3}×
223^{3} 
3^{15} ×11^{3}
× 97^{5} ×409 
5^{15} × 179^{4}
× 2141 
1.4123 
F.R. 
December 25, 2008 
2× 5^{10}× 13^{4} 
3^{15} ×7× 31^{7}
× 45817 
11^{8} × 109^{2}
× 3677^{3} 
1.4232 
F.R. 
December 20, 2008 
23^{3} × 53^{6}×
3167^{2} 
2^{8} × 3^{29}
× 11399^{2} 
5^{7} × 7^{4}
× 13^{12}× 523 
1.4501 
F.R. 
May 09, 2008 
23^{8} × 37^{4} 
2^{28} × 3^{7}
×11^{4} ×19^{3}
× 61 ×127×173^{2} 
5^{18} × 17^{4}
× 43^{2}× 4817^{2}

1.4502 
I.C. 
December 20, 2007 
5^{2} × 13^{4} 
17^{4} × 141971 
3^{18} × 7^{4}
× 11^{3}× 89^{4}

1.4226 
F.R. 
September 20, 2007 
5^{2} × 23^{10}×
106531 
7^{11} × 11^{3}
× 193^{4} 
2^{4} × 3^{19}
× 17^{8}× 29 
1.4646 
F.R. 
September 07, 2007 
2^{24} × 5^{5}×
47^{5} × 181^{2} 
13^{14} × 19×
103 × 571^{2}×
4261 
7^{28} × 17× 37^{2}

1.447420 
F.R. 
Auguste 26, 2007 
2^{8} × 47×
16421 
5^{12} × 439^{6}

2^{59} × 41×
73939 
1.4017 
F.R. 
August 25, 2007 
5^{9} × 17^{2}×
23^{4} × 37^{2}×
43× 4817 
3^{14} × 11^{8}
× 61^{2} × 74^{4} 
2^{52} × 19^{6}
× 127^{2} 
1.419184 
F.R. 
August 19, 2007 
11^{6} × 23^{3}×
449^{2} 
2^{26} × 3^{10}
× 13 × 17^{2} × 263^{2} 
5^{3} × 7^{4}
×
19^{3}× 29^{8} 
1.411854 
F.R. 
August 16, 2007 
3^{10} × 7^{6}×
541× 22031 
5^{3} × 29^{6}
× 1013^{4} 
2^{17} × 11^{16}
× 13 
1.428912 
F.R. 
August 14, 2007 
5^{11} × 13^{2} 
2^{8} × 17^{6}
× 23^{5} × 149 
3^{7} × 7^{6}
×
11× 29^{3}× 293^{2}

1.425182 
F.R. 
Auguste 07, 2007 
3^{21} × 7×
4498001 
5^{10} × 499^{5}

2^{28} × 17^{3}
× 47^{5} 
1.4051 
F.R. 
August 03, 2007 
31^{3} × 61^{5} 
17^{10} × 83^{2}
× 41059619 
2 × 3^{3} × 5^{17}
× 7^{12} 
1.451917 
F.R. 
August 03, 2007 
3^{6} × 47^{7}×
167 
7^{9} × 11^{4}
× 23^{4} × 68473 
2^{5} × 5^{15}
× 103^{5} 
1.425728 
F.R. 
May 16, 2007 
2^{13} 7^{4}×
653^{2} 
3^{18} × 5^{5}
× 181× 673^{2} 
11 × 13^{13} ×
31^{3} 
1.441775 
Reken
mee met ABC, abcathome

March 19, 2007 
2^{7} × 89^{2} 
5^{4} × 7^{6}
×
11^{2}× 71^{4} 
3^{13} × 19^{3}
× 4547^{2} 
1.4342 
J.D.B.S.H.W.L.W.J.P. 
March 19, 2007 
2^{32} × 73^{3} 
3^{14} × 5^{3}
× 11× 13^{5}× 557 
7^{13} × 23^{2}
× 163^{2} 
1.4323 
J.D.B.S.H.W.L.W.J.P. 
March 19, 2007 
1 
3^{7} × 7^{5}
×
13^{5}× 17× 1831 
2^{30} × 5^{2}
× 127× 353^{2} 
1.4012 
J.D.B.S.H.W.L.W.J.P. 
March 07, 2007 
2^{47} × 97 
5^{5} × 7^{8}
×
89× 739^{2} 
3^{17} × 11^{6}
× 13^{2} × 23 
1.419559 
Reken
mee met ABC, abcathome

Table V. Largest good abcexamples,
sorted by number of digits
log(c) = (a,b,c) = ^{_____________________}, log(rad(abc))

Authors:
F.R. : Frank Rubin T.D. : Tim Dokchitser

Number of digits of c 
a 
b 
c 

Author 
30 
2^{14}× 3^{6}×
42487^{3} 
5^{14} ×29^{12}
×83 
7^{8} × 11^{3}
× 47^{7} × 4610911 
1.4126 
F.R. 
30 
2^{37}× 3^{12}×2109^{3} 
5^{13} ×13^{15}
×2939 
7^{23} × 11×
793345871 
1.4121 
F.R. 
29 
23^{8} × 37^{4} 
2^{28} × 3^{7}
×11^{4} ×19^{3}
× 61 ×127×173^{2} 
5^{18} × 17^{4}
× 43^{2}× 4817^{2}

1.4502 
I.C. 
29 
2^{24} × 5^{5}×
47^{5} × 181^{2} 
13^{14} × 19×
103 × 571^{2}×
4261 
7^{28} × 17× 37^{2}

1.447420 
F.R. 
28 
5^{9} × 17^{2}×
23^{4} × 37^{2}×
43× 4817 
3^{14} × 11^{8}
× 61^{2} × 74^{4} 
2^{52} × 19^{6}
× 127^{2} 
1.419184 
F.R. 
28 
31^{2}× 61^{7}×
3889 
3^{23} ×11^{7}
× 151^{3} ×173 
2^{23} × 5^{6}
× 7^{3}× 83^{3}×
349^{3} 
1.4150 
J.W. 
To Index
 [Ba] Baker, A.
Logarithmic forms and the $abc$conjecture. Gyoery,
Kalman (ed.) et al., Number theory. Diophantine,
computational and
algebraic aspects. Proceedings of the international conference, Eger,
Hungary, July 29August 2, 1996. Berlin: de Gruyter. 3744 (1998).
 [Ba2] Baker, A.
Experiments on the $abc$conjecture.
Publ. Math. 65, No. 34, 253260 (2004).
 [BalLanShoWal] Balasubramanian, R.;
Langevin, M.; Shorey, T.N.; Waldschmidt, M. On
the maximal length of two sequences of integers in
arithmetic progressions with the same prime divisors.
Monatsh. Math. 121, No.4, 295307 (1996).
 [BayTe1] Bayat, M.; Teimoori, H.
A generalization of Mason's theorem for four
polynomials. Elem. Math. 59, No.1, 2328 (2004).
 [BayTe2] Bayat, M.; Teimoori, H.
A new bound for an extension of Mason's theorem for
functions of several
variables. Arch. Math. 82, No.3, 230239 (2004).
 [Bo] Bombieri, E. Roth's
theorem and the abc conjecture.
preprint (1994).
 [BoMu] Bombieri, E.; Mueller, J.
The generalized Fermat equation in function fields.
J. Number Theory 39, No.3, 339350 (1991).
 [BoMu] Bombieri, E.; Mueller, J.
On a conjecture of Siegel. Monatsh. Math.
125 no. 4, 293308 (1998).
 [Bor] Borisov, Alexandr
On some polynomials allegedly related to the $abc$
conjecture.
Acta Arith. 84, No.2, 109128 (1998).
 [Bro] Broberg, Niklas Some
examples related to the $abc$conjecture for
algebraic number fields.
Math. Comput. 69, No.232, 17071710 (2000).
 [Broug] Broughan, Kevin
Relaxations of the ABC conjecture using integer k'th
roots.
New Zealand Journal of Mathematics, vol 35, 121136 (2006).
 [Br1] Browkin, J. The $abc$
conjecture.
Bambah, R. P. (ed.) et al., Number theory. Basel: Birkhaeuser. Trends
in Mathematics. 75105 (2000).
 [Br2] Browkin, J. A
consequence of an effective form of the
$abc$conjecture.
Colloq. Math. 82, No.1, 7984 (1999).
 [Br3] Browkin, J. The
$abc$conjecture for algebraic numbers.
Acta Math. Sinica. (To appear).
 [BrBrz] Browkin, J.; Brzezinski, J.
Some remarks on the $abc$conjecture. Math.
Comput. 62, No.206, 931939 (1994).
 [BrFiGrSc] Browkin, J.; Filaseta, M.; Greaves,
G.; Schinzel, A. Squarefree values of polynomials
and the $abc$conjecture. Greaves, G. R.
H. (ed.) et al., Sieve methods,
exponential sums, and their
applications in number theory. Proceedings of a symposium, Cardiff, UK,
July 1721, 1995. Cambridge: Cambridge University Press. Lond. Math.
Soc. Lect. Note Ser. 237, 6585 (1997).
 [Brz] Brzezinski, Juliusz
ABC on the $abc$conjecture.
Normat 42, No.3, 97107 (1994).
 [Bu] Buium, Alexandru The $abc$
theorem for abelian
varieties.
Int. Math. Res. Not. 1994, No.5, 219233 (1994).
 [CoDr] Cochrane, Todd; Dressler, Robert E.
Gaps between integers with the same prime factors.
[J] Math. Comput. 68, No.225, 395401 (1999).
 [Coh] Cohn, J. H. E. The
Diophantine equation $(a\sp n1)(b\sp n1)=x\sp 2$.
Period. Math. Hungar. 44 (2002), no. 2, 169175
 [Cor] Cornelissen, Gunther
Stockage diophantien et hypothese $abc$
generalisee. (Diophantine
storing and the generalized $abc$hypothesis).
C. R. Acad. Sci., Paris, Ser. I, Math. 328, No.1, 38 (1999).
 [Corv] Corvaja, P. An
explicit version of the theorem of RothRidout.
Rend. Semin. Mat., Torino 53, No.3, 253260 (1995).
 [CrLiZh] Croot, Ernie; Li, RenCang; Zhu, Hui
June The $abc$ conjecture and correctly rounded
reciprocal
square roots.
Theor. Comput. Sci. 315, No.23, 405417 (2004).
 [CuGrTu] Cutter, Pamela; Granville, Andrew;
Tucker, Thomas J. The number of fields generated
by the square root of
values of a given polynomial.
Canad. Math. Bull. 46 (2003), no. 1, 7179.
 [Da] Dabrowski, Andrzej On
the diophantine equation $x!+A=y\sp 2$.
Nieuw Arch. Wiskd., IV. Ser. 14, No.3, 321324 (1996).
 [DarGr] Darmon, Henri; Granville, Andrew.
On the equations $z\sp m = F(x,y)$ and $Ax\sp p + By\sp
q = Cz\sp r$. Bull. Lond. Math. Soc. 27, No.6, 513543
(1995).
 [Dav] Davies, Daniel A note
on the limit points associated with the
generalized
$abc$conjecture for $\bbfZ [t]$.
Colloq. Math. 71, No.2, 329333 (1996).
 [Do] Dokchitser, Tim LLL
and abc.
J. Number Theory 107, No.1, 161167 (2004).
 [Dub] Dubickas, Arturas On
a height related to the $abc$ conjecture.
Indian J. Pure Appl. Math. 34, No.6, 853857 (2003).
 [El] Elkies, Noam D. $abc$
implies Mordell.
Int. Math. Res. Not. 1991, No.7, 99109 (1991).
 [Ell] Ellenberg, Jordan S.
Congruence ABC implies ABC.
Indag. Math. N.S., 11 (2), 197200 (2000).
 [FiKo] Filaseta, Michael; Konyagin, Sergej
On a limit point associated with the $abc$conjecture.
Colloq. Math. 76, No.2, 265268 (1998).
 [Fr1] van Frankenhuysen, Machiel Good
abc examples over number fields. Preprint
 [Fr2] van Frankenhuysen, Machiel
A lower bound in the abc conjecture.
J. Number Theory. 82, 91ï¿½95 (2000).
 [Fr3] van Frankenhuysen, Machiel The
abc conjecture implies Roth's theorem and Mordell's
conjecture.
Math. Contemporanea, 76, 4572 (1999).
 [Fr4] van Frankenhuysen, Machiel
The abc conjecture implies Vojta's height inequality for
curves. J. Number Theory 95 (2002), no. 2, 289302.
 [Fre1] Frey, Gerhard
Elliptic curves and solutions of $AB=C$.
Theorie des nombres, Semin. Paris 1985/86, Prog. Math. 71, 3951 (1987).
 [Fre2] Frey, G. Links
between elliptic curves and solutions of $AB=C$.
J. Indian Math. Soc., New Ser. 51, 117145 (1987).
 [Fre3] Frey, Gerhard
Links between solutions of $AB=C$ and elliptic curves.
Number theory, Proc. 15th Journ. Arith., Ulm/FRG 1987, Lect. Notes
Math.
1380, 3162 (1989).
 [Fre4] Frey, Gerhard Galois
representations attached to elliptic curves and
Diophantine problems.
Jutila, Matti (ed.) et al., Number theory. Proceedings of the Turku
symposium on number
theory in memory of Kustaa Inkeri, Turku, Finland, May 31June 4, 1999.
Berlin: de Gruyter.
71104 (2001).
 [Go] Goldfeld, Dorian
Modular Forms, Elliptic Curves and the $ABC$Conjecture. A
panorama of number theory or the view from Baker's
garden (Zï¿½rich, 1999), 128147,
Cambridge Univ. Press, Cambridge, 2002.
 [GoSz] Goldfeld, Dorian; Szpiro, Lucien
Bounds for the order of the TateShafarevich group.
[J] Compos. Math. 97, No.12, 7187 (1995).
 [GrSt] Granville, Andrew; Stark, H.M.
ABC implies no `Siegel zero' for $L$functions of
characters with negative discriminant. Invent. Math. 139,
no. 3, 509523 (2000).
 [Gr1] Granville, Andrew
Some conjectures related to Fermat's Last Theorem.
Number theory, Proc. 1st Conf. Can. Number Theory Assoc., Banff/Alberta
(Can.) 1988, 177192 (1990).
 [Gr2] Granville, Andrew On
the number of solutions to the generalized Fermat
equation.
Dilcher, Karl (ed.), Number theory. Fourth conference of the Canadian
Number Theory Association, July 28, 1994, Dalhousie University,
Halifax,
Nova Scotia, Canada. Providence, RI: American Mathematical Society. CMS
Conf. Proc. 15, 197207 (1995).
 [Gr3] Granville, Andrew
ABC allows us to count squarefrees.
Int. Math. Res. Not. 1998, No.19, 9911009 (1998).
 [GrTu]
Granville, Andrew; Tucker, Thomas J. It's as
easy as $abc$. Notices Amer. Math.
Soc. 49 (2002), no. 10, 12241231.
 [GreNi] Greaves, George; Nitaj, Abderrahmane
Some polynomial identities related to the abcconjecture.
Gyoery, Kalman (ed.) et al., Number theory in progress.
Proceedings of the international
conference organized by the Stefan Banach International Mathematical
Center in honor of
the 60th birthday of Andrzej Schinzel, Zakopane, Poland, June 30July
9, 1997. Volume 1: Diophantine problems and polynomials. Berlin: de
Gruyter. 229236 (1999).
 [Gy] Gyory, Kalman On the
abc conjecture in algebraic number fields.
Acta Arithmetica. To appear.
 [He] Hellegouarch, Yves
Analogues en caracteristique $p$ d'un theoreme de Mason.
(Two
$p$analogs for a theorem of Mason).
C. R. Acad. Sci., Paris, Ser. I, Math. 325, No.2, 141144 (1997).
 [HSi] Hindry, Marc; Silverman, Joseph
The canonical height and integral points on elliptic
curves.
Invent. Math. 93 (1988), 419450.
 [HuYa] Hu, PeiChu; Yang, ChungChun
The ''$abc$'' conjecture over function fields.
Proc. Japan Acad., Ser. A 76, No.7, 118120 (2000).
 [HuYa1] Hu, PeiChu; Yang, ChungChun
Notes on a generalized $abc$conjecture over function
fields.
Ann. Math. Blaise Pascal 8, No.1, 6171 (2001).
 [HuYa2] Hu, PeiChu; Yang, ChungChun A
note on the abc conjecture.
Comm. Pure Appl. Math. 55 (2002), no. 9, 10891103.
 [HuYa3] Hu, PeiChu; Yang, ChungChun
A generalized $abc$conjecture over function fields.
J. Number Theory 94, No.2, 286298 (2002).
 [HuYa4] Hu, PeiChu; Yang, ChungChun
Some progress in nonArchimedean analysis.
Contemp. Math. 303, 3750 (2002).
 [Ic] Ichimura, Humio A note
on Greenberg's conjecture and the abc
conjecture.
Proc. Am. Math. Soc. 126, No.5, 13151320 (1998).
 [Ka] Katayama, Shinichi
The $abc$ conjecture and the fundamental system of units
of certain real bicyclic biquadratic fields.
Proc. Japan Acad. Ser. A Math. Sci. 75, No.10, 198199 (1999).
 [Ki] Kim, Minhyong The ABC
inequalities for some moduli spaces of
loggeneral type. Math. Res. Lett. 5, no. 4, 517522
(1998).
 [La1] Lang, Serge Old and
new conjectured diophantine inequalities.
Bull. Am. Math. Soc., New Ser. 23, No.1, 3775 (1990).
 [La2] Lang, Serge
Number theory III: Diophantine geometry.
Encyclopaedia of Mathematical Sciences. 60. Berlin etc.:
SpringerVerlag.
xi, 296 p. (1991).
 [La3] Lang, Serge Die $abc$Vermutung.
(The $abc$conjecture). Elem. Math. 48,
No.3, 8999 (1993).
 [Lan1] Langevin, Michel
Partie sans facteur carre d'un produit d'entiers
voisins. (Squarefree
divisor of a product of neighbouring integers).
Approximations diophantiennes et nombres transcendants, C.R. Colloq.,
Luminy/ Fr. 1990, 203214 (1992).
 [Lan2] Langevin, M. Cas
d'egalite pour le theoreme de Mason et applications
de la conjecture $(abc)$.
(Extremal cases for Mason's theorem and applications of the $(abc)$
conjecture). C. R. Acad. Sci., Paris, Ser. I 317, No.5,
441444 (1993).
 [Lan3] Langevin, Michel
Sur quelques consequences de la conjecture $(abc)$
en arithmetique et en
logique. (On certain consequences of the $(abc)$
conjecture in arithmetic
and in logic).
Rocky Mt. J. Math. 26, No.3, 10311042 (1996).
 [Lan4] Langevin, Michel
Liens entre le theoreme de Mason et la conjecture $(abc)$.
(Connections
between Mason's theorem and the $(abc)$ conjecture).
Gupta, Rajiv (ed.) et al., Number theory. Fifth
conference of the
Canadian
Number Theory Association, Ottawa, Ontario, Canada, August 1722,
1996. Providence, RI: American Mathematical Society. CRM Proc. Lect.
Notes. 19, 187213 (1999).
 [Lan5] Langevin, Michel
Imbrications entre le theoreme de Mason, la descente de
Belyi et les
differentes formes de la conjecture $(abc)$. (Links between Mason's
theorem, Belyi descent and different versions of the
$(abc)$conjecture). J. Theor. Nombres Bordx. 11, No.1,
91109 (1999).
 [Lo] Lockhart, P. On the
discriminant of a hyperelliptic curve.
Trans. Am. Math. Soc. 342, No.2, 729852 (1994).
 [Lu] Luca, Florian On the
Diophantine equation $p^{x_1}  p^{x_2} = q^{y_1}
 q^{y_2}$. Indag. Math., New Ser. 14, No.2, 207222
(2003).
 [Ma] Mason, R.C. Diophantine
Equations Over Function Fields.
London Mathematical Society Lecture Note Series, 96. Cambridge etc.:
Cambridge University Press. (1984).
 [Mas1] Masser, D.W. Note on
a conjecture of Szpiro.
Les pinceaux de courbes elliptiques, Semin., Paris/Fr. 1988, Asterisque
183, 1923 (1990).
 [Mas2] Masser, D.W. On
$abc$ and discriminants.
Proc. Am. Math. Soc. 130, No.11, 31413150 (2002).
 [Mo] MoretBailly, Laurent
Hauteurs et classes de Chern sur les surfaces
arithmetiques. (Heights and
Chern classes on arithmetic surfaces).
Les pinceaux de courbes elliptiques, Semin., Paris/Fr. 1988, Asterisque
183,
3758 (1990).
 [Mu] Mueller, Julia The $abc$inequality
and the
generalized Fermat equation in function
fields.
Acta Arith. 64, No.1, 718 (1993).
 [Mur] Murty, M.Ram The
ABC conjecture and exponents of class groups of
quadratic fields.
Murty, V. Kumar (ed.) et al., Number theory. Proceedings of the
international conference on discrete mathematics and number theory,
Tiruchirapalli, India, January 36, 1996 on the occasion of the 10th
anniversary of the Ramanujan Mathematical Society. Providence, RI:
American Mathematical Society. Contemp. Math. 210, 8595 (1998).
 [MurWo] Murty, Ram; Wong, Siman
The $ABC$ conjecture and prime divisors of the Lucas and
Lehmer sequences. Number theory for the millennium, III
(Urbana, IL, 2000), 4354, A K Peters, Natick, MA, 2002.
 [Ni1] Nitaj, Abderrahmane
An algorithm for finding good $abc$examples. C.
R. Acad. Sci., Paris, Ser. I 317, No.9, 811815
(1993).
 [Ni2] Nitaj, Abderrahmane
Algorithms for finding good examples for the $abc$
and Szpiro
conjectures.
Exp. Math. 2, No.3, 223230 (1993).
 [Ni3] Nitaj, Abderrahmane
Aspects experimentaux de la conjecture $abc$.
(Experimental aspects of
the $abc$conjecture).
David, Sinnou (ed.), Number theory. Seminaire de Theorie des Nombres de
Paris 199394. Cambridge: Cambridge University Press. Lond. Math. Soc.
Lect. Note Ser. 235, 145156 (1996).
 [Ni4] Nitaj, Abderrahmane
La conjecture $abc$. (The $abc$
conjecture).
Enseign. Math., II. Ser. 42, No.12, 324 (1996).
 [Oe] Oesterle. J.
Nouvelles approches du ``theoreme'' de Fermat. (New
approaches to Fermat's last theorem).
Semin. Bourbaki, 40eme Annee, Vol. 1987/88, Exp. No.694, Asterisque
161/162, 165186 (1988).
 [Ov] Overholt, Marius The
diophantine equation $n!+1=m\sp 2$.
Bull. Lond. Math. Soc. 25, No.2, 104 (1993).
 [Ri1] Ribenboim, Paulo
$ABC$ candies.
J. Number Theory 81, No.1, 4860 (2000).
 [Ri2] Ribenboim, Paulo
The (ABC) Conjecture and the radical index of integers.
Acta Arith. 96, No.4, 389404 (2001).
 [Ri3] Ribenboim, Paulo On
square factors of terms of binary recurring sequences
and the $ABC$ conjecture.
Publ. Math. 59, No.34, 459469 (2001).
.
 [RiWa] Ribenboim, Paulo; Walsh, Gary
The ABC conjecture and the powerful part of terms in
binary recurring
sequences.
J. Number Theory 74, No.1, 134147, (1999).
 [Sc] Scanlon, Thomas The $abc$
theorem for commutative
algebraic groups in characteristic
$p$.
Int. Math. Res. Not. 1997, No.18, 881898 (1997).
 [Si] Silverman, Joseph H.
Wieferich's criterion and the abcconjecture. J.
Number Theory 30, No.2, 226237 (1988).
 [Si2] Silverman, Joseph H.
A quantitative version of Siegel's theorem: Integral
points on elliptic curves and Catalan curves.
J. Reine Angew. Math. 378 (1987), 60100.
 [Sch] Schmidt, Wolfgang M.
Diophantine approximations and diophantine equations. Lecture Notes in
Mathematics. 1467. Berlin etc.: SpringerVerlag. viii, 217 p. (1991).
 [Sh] Shapiro, Harold N.; Sparer, Gerson H.
Extension of a theorem of Mason.
Commun. Pure Appl. Math. 47, No.5, 711718 (1994).
 [Sm] Smirnov, A.L.
Hurwitz inequalities for number fields.
St. Petersbg. Math. J. 4, No.2, 357375 (1993); translation from
Algebra
Anal. 4, No.2, 186209 (1992).
 [Sny] Snyder, Noah An
alternate proof of Mason's theorem.
Elem. Math. 55, No.3, 9394 (2000).
 [SteTi] Stewart, C.L.; Tijdeman, R. On
the OesterleMasser conjecture.
Monatsh. Math. 102, 251257 (1986).
 [SteYu1] Stewart, C.L.; Yu, Kunrui
On the $abc$ conjecture.
Math. Ann. 291, No.2, 225230 (1991).
 [SteYu2] Stewart, C.L.; Yu, Kunrui
On the $abc$ conjecture. II
Duke Math. J. 108 No.1 169181 (2001).
 [Sto] Stothers, W.W. Polynomial
identities and Hauptmoduln. [J] Q.
J. Math., Oxf. II. Ser. 32, 349370 (1981).
 [Ti1] Tijdeman, R.
Diophantine equations and diophantine approximations.
Number theory and applications, Proc. NATO ASI, Banff/Can. 1988, NATO
ASI Ser., Ser. C 265,
215243 (1989).
 [Ti2] Tijdeman, R.
Riemann's Hypothesis and the ABCconjecture.
Summer course 1999: unproven conjectures. Amsterdam: Stichting
Mathematisch Centrum,
Centrum voor Wiskunde en Informatica. CWI Syllabus. 45, 3144 (1999).
 [Va] Vaserstein, L.
Quantum (abc)theorems.
J. Number Theory 81, No.2 , 351368 (2000).
 [Vo1] Vojta, Paul
Diophantine Approximations and Value Distribution Theory. Lecture Notes
in Mathematics, 1239. Berlin etc.: SpringerVerlag. X, 132 p. (1987).
 [Vo2] Vojta, Paul A more
general $abc$ conjecture.
Int. Math. Res. Not. 1998, No.21, 11031116 (1998).
 [Vo3] Vojta, Paul On the $abc$
conjecture and
diophantine approximation by rational points.
Am. J. Math. 122, No.4, 843872 (2000).
 [Vo4] Vojta, Paul
Correction to ``On the $abc$
conjecture and diophantine approximation by rational points''.
Am. J. Math. 123, No.2, 383384 (2001).
 [VuDo] Vu Hoai An; Doan Quang Manh
The ``abc'' conjecture for $p$adic functions of several
variables.
Southeast Asian Bull. Math. 27, No.6, 959972 (2004).
 [Wa1] Walsh, P.G. On
integer solutions to $x\sp 2dy\sp 2=1, z\sp 22dy\sp
2=1$.
Acta Arith. 82, No.1, 6976 (1997).
 [Wa2] Walsh, P.G. On a
conjecture of Schinzel and Tijdeman.
Gyoery, Kalman (ed.) et al., Number theory in progress. Proceedings
of the international conference organized by the Stefan Banach
International
Mathematical Center in honor of the 60th birthday of Andrzej Schinzel,
Zakopane, Poland, June 30July 9, 1997. Volume 1: Diophantine
problems and polynomials. Berlin: de Gruyter. 577582 (1999).
 [Wal1] Waldschmidt, M.
Open diophantine problems.
Moscow Mathematical Journal
Volume 4, Number 1, January–March 2004, Pages
245–305.
 [Wal2] Waldschmidt, M.
Perfect Powers:
Pillai’s works and their developments, In:
Collected works of S. Sivasankaranarayana Plliai, Eds. R.
Balasubramanian and R. Thangadurai, Collected Works Series, no. 1,
Ramanujan Mathematical Society, Mysore, 2010.
 [Wan] Wang, Julie TzuYueh,
A note on Wronskians and the ABC theorem in function
fields of prime
characteristic.
Manuscr. Math. 98, No.2, 255264 (1999).
 [We1] de Weger, B.M.M. Algorithms
for diophantine equations. CWI Tract, 65.
Amsterdam: Centrum
voor Wiskunde en Informatica. viii,
212 p. Dfl. 33.00 (1989).
 [We2] de Weger, Benjamin
M.M. $A+B=C$ and big Sha's.
Q. J. Math., Oxf. II. Ser. 49, No.193, 105128 (1998).
 [Za] Zannier, Umberto On
Davenport's bound for the degree of $f\sp 3  g\sp 2$
and Riemann's
existence theorem.
Acta Arith. 71, No.2, 107137 (1995); Correction 74, No.4, 387 (1996).
To Index
 On
the abc Conjecture
and some of its consequences , M. Waldschmidt, 6th World
Conference on 21st Century
Mathematics, 2013
 Open
Diophantine Problems, M. Waldschmidt, Moscow Mathematical
Journal
Volume 4, Number 1, January–March 2004, Pages
245–305.
 Perfect
Powers:
Pillai’s works and their developments, , M.
Waldschmidt
 It's
as
easy as the abc conjecture, A. Granville, Notices of the AMS,
Volume
49, Number 10
 Questions
about Powers of Numbers, B. Mazur, Notices of the AMS, Volume
47, Number
2
 LLL
& ABC , T. Dokchister.
 The
abcconjecture is true
for at least $N(c), 1 \leq N(c) <\phi(c)/2$, partitions a, b of c,
C.M. Petridi, preprint
 The
abcconjecture for
algebraic number fields.,
J. Browkin.
To Index
 Méthodes
de transcendance et géométrie diophantienne,
A. Surroca,
Ph.D. Thesis, Université de Paris 6, 2003
 The
abc conjecture,
Jeffrey Paul Wheeler, Master Thesis, University of Tennessee, Knoxville
 La
conjecture
abc
et ses aplications, J.N. Fournier, Master Thesis,
Université
Laval, Canada, 1999
 Conséquences
et
aspects expérimentaux des conjectures abc et de Szpiro,
A.
Nitaj,
Ph.D. Thesis, Université de Caen, 1994
 Hyperbolic
spaces and the
abc conjecture, M. van Frankenhuysen, Ph.D. Thesis,
Universiteit Nijmegen,
1995
 Generalization
of the ABCconjecture,
N.R. Bruin, Master Thesis, Leiden University, 1995
 The ABC
Conjecture,
Keldon Drudge, M.Sc. Thesis, McGill University, 1995.
 The Wieferich criterion, the
ABC conjecture and
Shimura's
correspondence,
Satya Mohit, M.Sc. Thesis, Queen's University, 1998.
 Powerful
Numbers, the
abc Conjecture, and RamanujanNagell Equations, Karl Kihm
Oman, Ph.D.
Thesis, Wayne State University, 1998
 Quelques
applications de la conjecture abc, Stéphane
Fishler, Mémoire
de DEA, Paris, 1998
To Index
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