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 July 05, 2008

Index

The abc conjecture
rad(n): For a natural number, let rad(n) be the product of all distinct prime divisors of n. E.g. if n=25 × 37 × 11 × 172 then rad(n)=2 × 3 × 11 × 17=1122.
The abc conjecture: 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.

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Generalizations

  1. The abc theorem for polynomials. For a polynomial P with complex coefficients let N0=N0(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
    2. max(deg(A), deg(B), deg(C))  N0(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))  N0(ABC)-2.
  2. The abc conjecture for binary forms. It is shown in [Lan2] that the abc conjecture implies the following conjecture.

    3. 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.
  3. The n-term abc conjecture for integers. In 1994, Browkin and Brzezinski [Br-Brz] proposed the following conjecture.

    4. Given any integer n > 2 and any > 0, there exists a constant Cn, such that for all integers a1, ..., an with a1+...+ an=0, gcd( a1,..., an)=1 and no proper zero subsum, we have
    max(|a1|,...,|an|)  Cn,(rad(a1 × ... × an))2n-5+.
  4. 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.

    5. 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.
  5. The abc conjecture for number fields. Let K be an algebraic number field and let VK denote the set of primes on K, that is, any v in VK is an equivalence class of non-trivial norms on K (finite or infinite). Let ||x||v=NK/Q(P)-vP(x) if v is a prime definied by a prime ideal P of the ring of integers OK in K and vP is the corresponding valuation, where NK/Q is the absolute norm. Let ||x||v=|g(x)|e for all non-conjugate 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
    6. HK(a,b,c) = v in VK max(||a||v, ||b||v, ||c||v),
    and the radical (or conductor) of (a,b,c) by
    radK(a,b,c) =  P in IK(a,b,c)NK/Q(P),
    where IK(a,b,c) is the set of all prime ideals P of OK for which ||a||v, ||b||v , ||c||v are not equal. Let DK/Q denote the discriminant of K. K. Gyory new results on the uniform abc conjecture for number fields:
  6. The abc theorem for non-archimedean meromorphic function fields.  Let K be a non-archimedean 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 [Hu-Ya] 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 [Hu-Ya] and [Hu-Ya3]). Stothers-Mason's abc theorem for polynomials is an application of this result.
  7. The k-term abc theorem for non-archimedean meromorphic function fields.  Let K be a non-archimedean algebraically closed field of characteristic zero. Let fj(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 f0+f1+....+fk = 0. In 2002, Hu and Yang [Hu-Ya3] showed that
    max{T(r,fj)} < N(r,1/f0, 1/f1,..., 1/fk))-k(k-1)log(r)/2+O(1),
    where T and N are functions related to Nevanlinna's value distribution theory (see [Hu-Ya3]). Stothers-Mason's abc theorem for polynomials is an application of this result with k=2.
  8. Hu-Yang's k-term abc conjecture for integers.  Let a be a nonzero integer with the factorization |a|=p1i1...pnin where p1,...,pn are distinct primes. Define the k-radical of a to be
    rk(a)=pj|a pjmin(ij,k).
    In 2002, Hu and Yang [Hu-Ya3] proposed the following conjecture.
    Let ai, i=0...k, be nonzero integers without common factor and no proper subsum is equal to 0 such that
    a0+.....+ak =0.
    Then for  >0, there exists a constant C(k,) such that
    max|ai| < C(k,)R(a0...ak)1+,
    where
    R(a0...ak) = i rk-1(ai).
     If k=2, this corresponds to the abc conjecture.
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Consequences

Pierre de  Fermat
  1. Fermat's Last Theorem. Fermat's conjecture, known as Fermat's Last Theorem states that The equation xn+yn=zn 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 xn+yn=zn with gcd(x,y,z)=1 and n> 3.
    2. Andrew  Wiles
  2. The generalized Fermat equation. The abc conjecture implies [Ni4, Ti] that for given positive integers A, B, C, the generalized Fermat equation Axr+Bys=Czt 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 [Dar-Gr] proved that if r, s, t are fixed with 1/r+1/s+1/t< 1 then the equation Axr+Bys=Czt has only finitely many solutions in pairwise coprime integers x,y,z.
  3. Wieferich primes. A prime p is called a Wieferich prime if p2 divides 2 p-1-1. 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.

    4. Given a positive integer a> 1. There exist infinitely many primes p such that p2 does not divide ap-1-1.
  4. The Erdös-Woods 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 Erdös-Woods conjecture with k=3 except perhaps a finite number of counter examples.
  5. 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,...,K-1. This somewhere extends the Erdös-Woods conjecture to arithmetic progressions. It is shown in [Bal-Lan-Sho-Wal] 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.
  6. Hall's conjecture. The abc conjecture implies the following weak form of the Hall conjecture [Ni4, Sch]:

    7. Given any > 0, there exists a constant C> 0 such that for every positive integers x and y such that x3 - y2 is non-zero then |x3 - y2|> C max(x3, y2)1/6-.
  7. Erdös' conjecture on consecutive powerful numbers. A positive integer n is powerful if for every prime p dividing n, p2 also divides n. Every powerful number can be written as a2b3 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.
  8. Brown Numbers and Brocard's Problem. Pairs of numbers (m,n) satisfying Brocrad's Problem n!+1=m2 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=ym [Ni4] and x!+B2=y2 for general B [Da].
  9. 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 || < CN6+.
    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 Frey-Hellegouarch curves.
  10. 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]).
  11. 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 [Br-Fi-Gr-Sc] proved that the abc conjecture gives a positive answer for cyclotomic polynomials n(X) and for (Xn-1)/(X-1).
  12. Roth's theorem. In 1955, Klaus Roth proved that for every algebraic number  , the diophantine equation |-p/q| < 1/q2+, 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/q2+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]).
  13. 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 [Co-Dr] 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
    14. b-a> C a1/2-.
  14. Siegel zeros. Let L(s,) be the Dirichlet L-function 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 1-c/(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 [Gr-St] proved that the uniform abc conjecture for number fields implies that
    h(-d) > ( /3+o(1))d1/2(log d)-1(1/a),
    where the sum  runs over all reduced quadratic forms ax2+bxy+cy2 of discriminant -d. It is known since Mahler that if this holds, then the Dirichlet L-function L(s,(-d/.)) has no Siegel zero. Consequently, "ABC implies no Siegel Zero".
  15. Power free-values of polynomials. Langevin noted in [Lan2] the following conjecture which is a consequence of the abc conjecture.

    16. 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)).
  16. Counting squarefree-values 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(cfMN) pairs of positive integers m < M, n < N for which f(m,n)/B' is squarefree as M, N tend to infinity, where cf is a constant which depends only on f. A similar result exists for a polynomial f(x) with integer coefficients and no repeated roots.
  17. Bounds for the order of the Tate-Shafarevich group. Let E be an elliptic curve over Q with Tate-Shafarevich goup  and conductor N. The conjecture of Goldfeld and Zspiro asserts that for every  > 0, there exists a positive constant C such that || N1/2 + . Assuming the Birch and Swinnerton-Dyer conjecture, it is shown in [Go-Sz] that this conjecture is equivalent to the Szpiro conjecture for modular elliptic curves.
  18. 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 one-dimensional.
  19. 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=P2-4Q > 0 and gcd(P,Q)=1. Let U0=0, U1=1 and V0=2, V1=P. For each n > 1, let Un=PUn-1-QUn-2 and Vn=PVn-1-QVn-2. Ribenboim and Walsh [Ri-Wa] proved that the abc conjecture implies that the sets {n > 0, w(Un) > Un} and {n > 0, w(Vn) > Vn} are finite. It follows that Un and Vn are powerful for only finitely many terms.
  20. 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 Zp-extension of K. The pseudo-null 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.
  21. Exponents of class groups of quadratic fields. Given positive numbers g and x. Let N(x) be the number of quadratic number fields Q(d1/2) 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 xk+ where k=1/g if d < 0 and k=1/2g if d > 0.
  22. 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 [Gre-Ni] 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 [Fi-Ko] 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.
  23. Fundamental units of certain quadratic and biquadratic fields: added to this page on June 23, 2000. For a positive integer M let N1 = (M+(M2+/-4)1/2)/2, and N'1 = (M-(M2+/-4)1/2)/2. For any positive integer n, put gn = N1n+N'1n, hn = (N1n-N'1n)/(M2+/-4)1/2, N2=h2n+1+(h22n+1-1)1/2 and N3=g2n+1/M+(g22n+1/M2-1)1/2. Katayama [Ka] showed that if the abc conjecture is valid, then N2 is the fundamental unit of the real quadratic field Q((h22n+1-1)1/2), N3 is the fundamental unit of the real quadratic field Q((g22n+1/M2-1)1/2), and {N1, N2, N3} is a fundamental system of units of the real bicyclic biquadratic field Q((M2+/-4)1/2, (h22n+1-1)1/2) except for finitely many integers n.
  24. The Schinzel-Tijdeman 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)=y2z3 has only finitely many non-trivial solutions in integers x, y, z. Walsh [Wa2] proved that the abc conjecture implies this conjecture.
  25. 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 non-torsion point of the Mordell-Weil group E(K), then H(P)>C(K)log |NK/QDE/K| where DE/K is the minimal discriminant of E/K, H(P) is the canonical height of P and NK/Q is the norm. This conjecture of Lang follows from the ABC conjecture (see [H-Si]). To be more precise, Hindry and Silverman proved that H(P)>(20sz )-8[K:Q]10-1.1-4szlog |NK/QDE/K| where sz is the Szpiro ratio 
                log|NK/QDE/K|
         sz = ____________________.
               log|NK/QFE/K|
    and FE/K is the conductor of E/K.
  26. 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 S-integral 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.
  27. 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 [Cr-Li-Zh] showed that, assuming the abc conjecture, the number of the leading bits could be reduced to 2p.
  28. The abc-(k,m) conjecture for integers : added on May 23, 2005.  Let k >1 and n >0 be integers with the factorization n=p1i1...pnin where p1,...,pn are distinct primes. Define the k'th radical of n to be
    nk(n)=pjij||n pjUB(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)nk(abc)1+1/m.
     This conjecture is a wekened form of the abc conjecture.
  29. The diophantine equation pv-pw=qx-qy : added on May 23, 2005.  In 2003, Luca [Lu] showed that assuming the abc conjecture, the diophantine equation pv-pw=qx-qy has only finitely many positive integer solutions p, q, v,w,x,y where p and q are distinct prime numbers.
  30. The number of quadratic fields generated by a polynomial : added on May 23, 2005.  In 2003, Cutter, Granville an Tucker [Cu-Gr-Tu] 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.
To Index

Tables
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 abc-examples and good abc-Szpiro-examples.
For (nonzero) algebraic numbers a, b, c such that a + b = c, let K=Q(a/c) and 
                         log(HK(a,b,c))
     = (a,b,c) = ________________________________________.
                   log|DK/Q|+log(radK(a,b,c))
Triples satisfying  > 1.5 are called good algebraic abc-examples.

Authors of good abc-examples: 

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 abc-examples 
                       log(c)
      = (a,b,c) = _____________________,
                    log(rad(abc))
 
No a b c Author
1. 2 310 × 109 235 1.62991 E.R.
2. 112 32 × 56 × 73 221 × 23 1.62599 B.W.
3. 19 × 1307 7 × 292 × 318 28 × 322 × 54 1.62349 J.B-J.B
4. 283 511 × 132 28 × 38 × 173 1.58076 J.B-J.B, A.N 
5. 1 2 × 37 54 × 7 1.56789 B.W. 
6. 73 310 211 × 29 1. 54708 B.W. 
7. 72 × 412 × 3113 1116 × 132 × 79 2 × 33 × 523 × 953 1.54443 A.N. 
8. 53 29 × 317 × 132 115 × 17 × 313 × 137 1.53671 P.M-H.R
9. 13 × 196 230 × 5 313 × 112 × 31 1.52700 A.N. 
10. 318 × 23 × 2269 173 × 29 × 318 210 × 52 × 715 1.52216 A.N 
Rosenheinrich's HTML complete list of good abc-examples. 
de Smit's HTML complete list of good abc-examples. 
[PDF] [DVI] [PS] complete list of all known good abc-examples. 
[PDF] [DVI] [PS] T. Dokchitser's PDF, DVI or PS list of new good abc-examples. 

Table II. The top ten good abc-Szpiro-examples 
                     log(abc)
     = (a,b,c) = __________________________.
                   log(rad(abc))
 
No a b c Author
1. 13 × 196 230 × 5 313 × 112 × 31 4. 41901 A.N.
2. 25 × 112 × 199  515 × 372 × 47  37 × 711 × 743  4.26801  A.N. 
3. 219 × 13 × 103 711 311 × 53 × 112 4.24789 B.W.
4. 198 × 434 × 1492 215 × 523 × 101 313 × 13× 292 × 376 × 911 4.23181  T.D.
5. 235 × 72 × 172 × 19  327 × 1072  515 × 372 × 2311  4.23069  A.N. 
6. 318 × 23 × 2269  173 × 29 × 318 210 × 52 × 715  4.22979  A.N. 
7. 174 × 793 × 211  229 × 23 × 292 519 4.22960  A.N. 
8. 514 × 19  25 × 3 × 713 117 × 372 × 353  4.22532  A.N. 
9. 27 × 54 × 722 194 × 37× 474 × 536 314 × 11× 139 × 191 × 7829 4.21019  T.D.
10. 321 72 × 116 × 199 2 × 138 × 17 4.20094 A.N.
[PDF] [DVI] [PS] complete list of all known good abc-Szpiro examples. 
[PDF] [DVI] [PS] T. Dokchitser's PDF, DVI or PS list of new good abc-Szpiro examples. 

Table III. The top ten good purely algebraic abc-examples over K=Q(d) 
                         log(HK(a,b,c))
     = (a,b,c) = _____________________________________.
                   log|DK/Q|+log(radK(a,b,c)))
No Equation a b c w Author
1. w2-w-3=0 w (w+1)10(w-1) 29(w+1)5 (1+131/2)/2 2.029229 T.D.
2. wi3-2wi2+4wi-4=0, i=1,2,3 (w3-w2)w152 (w1-w3)w252 -(w2-w1)w352 de Weger's example  1.920859 B.W.
3. wi3+3wi-1=0, i=1,2,3 (2w1+1)(8w1-3)w116 (2w2+1)(8w2-3)w216 -(2w3+1)(8w3-3)w316 Dokchitser's example  1.918150 T.D.
4. w3+3w2-4w+1=0 (w2+4w-1)3 -(w2+4w-1)11(w2+3w-2)11 (w2+3w-2)8 Dokchitser's example  1.834740 T.D.
5. w2-2=0 w17 (1-w)5(3-w) (1+w)5(3+w) 21/2 1.768124 N.B.
6. w2-5w-4=0 -(2w+1)3(2w-13)2 (10w+7)10(w-6)17(-8w-7) (10w+7)7(w+1)5(2w-11)15 (5+411/2)/2 1.753452 T.D.
7. w3+3w+1=0 w14(w-2) (w2-w+1)5 -w2(w2+1)22 Dokchitser's example  1.751018 T.D.
8. w2-w-4=0 -(2w-5)4(w+1)5 (2w+3)4(w-2)5 35 (1+171/2)/2 1.712274 T.D.
9. w2-w-13=0 (w+3)4(w+1)3 -(w-4)4(w-2)3 3 × 76 (1+531/2)/2 1.719820 T.D.
10. w2-5w+2=0 (2w-1)8(w-4)5 -(w-1)5 35(2w-1)4 (5+171/2)/2 1.719820 T.D.
11. w2+w+2=0 (1-2w) (1-w)13 w13 (1+(-7)1/2)/2 1.707221 N.B.
12. w2-w-1=0 24× 32×(1-2w) w12 (1-w)12 1/2+51/2/2 1.697797 J.Bo.
... ... ... ... ... ... ... ...
.. w2-2=0 1 (1+w)14 132× (1+w)7w3 21/2 1.561437 N.B.
.. w2-7=0 (8-3w)2 (5-2w) (8-3w)7 (3-w)3 (5+2w)12 (4-3w)4 71/2 1.528940 N.B.
.. w2-6=0 72(5+2w)9 (2-w)9 (3-w) (1+w) (1-w) 1 (5+2w)8 61/2 1.518102 N.B.
Broberg's list of good algebraic abc-examples. 
Dokchitser's list of good algebraic abc-examples. 

Table IV. New good abc-examples, 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

Date a b c Author
September 7, 2007 224 × 55× 475 × 1812 1314 × 19× 103 × 5712× 4261 728 × 17× 372 1.447420 F.R.
August 25, 2007 59 × 172× 234 × 372× 43× 4817 314 × 118 × 612 × 744 252 × 196 × 1272 1.419184 F.R.
August 19, 2007 116 × 233× 4492 226 × 310 × 13 × 172 × 2632 53 × 74 × 193× 298 1.411854 F.R.
August 16, 2007 310 × 76× 541× 22031 53 × 296 × 10134 217 × 1116 × 13 1.428912 F.R.
August 14, 2007 511 × 132 28 × 176 × 235 × 149 37 × 76 × 11× 293× 2932 1.425182 F.R.
August 3, 2007 313 × 615 1710 × 832 × 41059619 2 × 33 × 517 × 712 1.451917 F.R.
August 3, 2007 36 × 477× 167 79 × 114 × 234 × 68473 25 × 515 × 1035 1.425728 F.R.
May 16, 2007 213 74× 6532 318 × 55 × 181× 6732 11 × 1313 × 313 1.441775 Reken mee met ABC, abcathome
March 19, 2007 27 × 892 54 × 76 × 112× 714 313 × 193 × 45472 1.4342 J.D.-B.S.-H.W.L.-W.J.P.
March 19, 2007 232 × 733 314 × 53 × 11× 135× 557 713 × 232 × 1632 1.4323 J.D.-B.S.-H.W.L.-W.J.P.
March 19, 2007 1 37 × 75 × 135× 17× 1831 230 × 52 × 127× 3532 1.4012 J.D.-B.S.-H.W.L.-W.J.P.
March 7, 2007 247 × 97 55 × 78 × 89× 7392 317 × 116 × 132 × 23 1.419559 Reken mee met ABC, abcathome

Table V. Largest good abc-examples, 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
29 224 × 55× 475 × 1812 1314 × 19× 103 × 5712× 4261 728 × 17× 372 1.447420 F.R.
28 59 × 172× 234 × 372× 43× 4817 314 × 118 × 612 × 744 252 × 196 × 1272 1.419184 F.R.
27 718× 2333 25× 518× 73× 173× 981439 338× 134× 5233 1.414570 T.D.
26 294× 22132 3× 132× 2312× 89× 14717 29× 516× 119× 79 1.412347 T.D.
To Index

Bibliography

  1. [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 29--August 2, 1996. Berlin: de Gruyter. 37-44 (1998).
  2. [Ba2] Baker, A. Experiments on the $abc$-conjecture. Publ. Math. 65, No. 3-4, 253-260 (2004).
  3. [Bal-Lan-Sho-Wal] 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, 295-307 (1996).
  4. [Bay-Te1] Bayat, M.; Teimoori, H. A generalization of Mason's theorem for four polynomials. Elem. Math. 59, No.1, 23-28 (2004).
  5. [Bay-Te2] Bayat, M.; Teimoori, H. A new bound for an extension of Mason's theorem for functions of several variables. Arch. Math. 82, No.3, 230-239 (2004).
  6. [Bo] Bombieri, E. Roth's theorem and the abc conjecture. preprint (1994).
  7. [Bo-Mu] Bombieri, E.; Mueller, J. The generalized Fermat equation in function fields. J. Number Theory 39, No.3, 339-350 (1991).
  8. [Bo-Mu] Bombieri, E.; Mueller, J. On a conjecture of Siegel. Monatsh. Math. 125 no. 4, 293--308 (1998).
  9. [Bor] Borisov, Alexandr On some polynomials allegedly related to the $abc$ conjecture. Acta Arith. 84, No.2, 109-128 (1998).
  10. [Bro] Broberg, Niklas Some examples related to the $abc$-conjecture for algebraic number fields. Math. Comput. 69, No.232, 1707-1710 (2000).
  11. [Broug] Broughan, Kevin Relaxations of the ABC conjecture using integer k'th roots. New Zealand Journal of Mathematics, vol 35, 121-136 (2006).
  12. [Br1] Browkin, J. The $abc$ conjecture. Bambah, R. P. (ed.) et al., Number theory. Basel: Birkhaeuser. Trends in Mathematics. 75-105 (2000).
  13. [Br2] Browkin, J. A consequence of an effective form of the $abc$-conjecture. Colloq. Math. 82, No.1, 79-84 (1999).
  14. [Br3] Browkin, J. The $abc$-conjecture for algebraic numbers. Acta Math. Sinica. (To appear).
  15. [Br-Brz] Browkin, J.; Brzezinski, J. Some remarks on the $abc$-conjecture. Math. Comput. 62, No.206, 931-939 (1994).
  16. [Br-Fi-Gr-Sc] 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 17--21, 1995. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 237, 65-85 (1997).
  17. [Brz] Brzezinski, Juliusz ABC on the $abc$-conjecture. Normat 42, No.3, 97-107 (1994).
  18. [Bu] Buium, Alexandru The $abc$ theorem for abelian varieties. Int. Math. Res. Not. 1994, No.5, 219-233 (1994).
  19. [Co-Dr] Cochrane, Todd; Dressler, Robert E. Gaps between integers with the same prime factors. [J] Math. Comput. 68, No.225, 395-401 (1999).
  20. [Coh] Cohn, J. H. E. The Diophantine equation $(a\sp n-1)(b\sp n-1)=x\sp 2$. Period. Math. Hungar. 44 (2002), no. 2, 169--175
  21. [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, 3-8 (1999).
  22. [Corv] Corvaja, P. An explicit version of the theorem of Roth-Ridout. Rend. Semin. Mat., Torino 53, No.3, 253-260 (1995).
  23. [Cr-Li-Zh] Croot, Ernie; Li, Ren-Cang; Zhu, Hui June The $abc$ conjecture and correctly rounded reciprocal square roots. Theor. Comput. Sci. 315, No.2-3, 405-417 (2004).
  24. [Cu-Gr-Tu] 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, 71--79.
  25. [Da] Dabrowski, Andrzej On the diophantine equation $x!+A=y\sp 2$. Nieuw Arch. Wiskd., IV. Ser. 14, No.3, 321-324 (1996).
  26. [Dar-Gr] 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, 513-543 (1995).
  27. [Dav] Davies, Daniel A note on the limit points associated with the generalized $abc$-conjecture for $\bbfZ [t]$. Colloq. Math. 71, No.2, 329-333 (1996).
  28. [Do] Dokchitser, Tim LLL and abc. J. Number Theory 107, No.1, 161-167 (2004).
  29. [Dub] Dubickas, Arturas On a height related to the $abc$ conjecture. Indian J. Pure Appl. Math. 34, No.6, 853-857 (2003).
  30. [El] Elkies, Noam D. $abc$ implies Mordell. Int. Math. Res. Not. 1991, No.7, 99-109 (1991).
  31. [Ell] Ellenberg, Jordan S. Congruence ABC implies ABC. Indag. Math. N.S., 11 (2), 197--200 (2000).
  32. [Fi-Ko] Filaseta, Michael; Konyagin, Sergej On a limit point associated with the $abc$-conjecture. Colloq. Math. 76, No.2, 265-268 (1998).
  33. [Fr1] van Frankenhuysen, Machiel Good abc examples over number fields. Preprint
  34. [Fr2] van Frankenhuysen, Machiel A lower bound in the abc conjecture. J. Number Theory. 82, 91�95 (2000).
  35. [Fr3] van Frankenhuysen, Machiel The abc conjecture implies Roth's theorem and Mordell's conjecture. Math. Contemporanea, 76, 45-72 (1999).
  36. [Fr4] van Frankenhuysen, Machiel The abc conjecture implies Vojta's height inequality for curves. J. Number Theory 95 (2002), no. 2, 289--302.
  37. [Fre1] Frey, Gerhard Elliptic curves and solutions of $A-B=C$. Theorie des nombres, Semin. Paris 1985/86, Prog. Math. 71, 39-51 (1987).
  38. [Fre2] Frey, G. Links between elliptic curves and solutions of $A-B=C$. J. Indian Math. Soc., New Ser. 51, 117-145 (1987).
  39. [Fre3] Frey, Gerhard Links between solutions of $A-B=C$ and elliptic curves. Number theory, Proc. 15th Journ. Arith., Ulm/FRG 1987, Lect. Notes Math. 1380, 31-62 (1989).
  40. [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 31-June 4, 1999. Berlin: de Gruyter. 71-104 (2001).
  41. [Go] Goldfeld, Dorian Modular Forms, Elliptic Curves and the $ABC$-Conjecture. A panorama of number theory or the view from Baker's garden (Z�rich, 1999), 128--147, Cambridge Univ. Press, Cambridge, 2002.
  42. [Go-Sz] Goldfeld, Dorian; Szpiro, Lucien Bounds for the order of the Tate-Shafarevich group. [J] Compos. Math. 97, No.1-2, 71-87 (1995).
  43. [Gr-St] Granville, Andrew; Stark, H.M. ABC implies no `Siegel zero' for $L$-functions of characters with negative discriminant. Invent. Math. 139, no. 3, 509--523 (2000).
  44. [Gr1] Granville, Andrew Some conjectures related to Fermat's Last Theorem. Number theory, Proc. 1st Conf. Can. Number Theory Assoc., Banff/Alberta (Can.) 1988, 177-192 (1990).
  45. [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 2-8, 1994, Dalhousie University, Halifax, Nova Scotia, Canada. Providence, RI: American Mathematical Society. CMS Conf. Proc. 15, 197-207 (1995).
  46. [Gr3] Granville, Andrew ABC allows us to count squarefrees. Int. Math. Res. Not. 1998, No.19, 991-1009 (1998).
  47. [Gr-Tu] Granville, Andrew; Tucker, Thomas J. It's as easy as $abc$. Notices Amer. Math. Soc. 49 (2002), no. 10, 1224--1231.
  48. [Gre-Ni] Greaves, George; Nitaj, Abderrahmane Some polynomial identities related to the abc-conjecture. 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 30--July 9, 1997. Volume 1: Diophantine problems and polynomials. Berlin: de Gruyter. 229-236 (1999).
  49. [Gy] Gyory, Kalman On the abc conjecture in algebraic number fields. Acta Arithmetica. To appear.
  50. [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, 141-144 (1997).
  51. [H-Si] Hindry, Marc; Silverman, Joseph The canonical height and integral points on elliptic curves. Invent. Math. 93 (1988), 419-450.
  52. [Hu-Ya] Hu, Pei-Chu; Yang, Chung-Chun The ''$abc$'' conjecture over function fields. Proc. Japan Acad., Ser. A 76, No.7, 118-120 (2000).
  53. [Hu-Ya1] Hu, Pei-Chu; Yang, Chung-Chun Notes on a generalized $abc$-conjecture over function fields. Ann. Math. Blaise Pascal 8, No.1, 61-71 (2001).
  54. [Hu-Ya2] Hu, Pei-Chu; Yang, Chung-Chun A note on the abc conjecture. Comm. Pure Appl. Math. 55 (2002), no. 9, 1089--1103.
  55. [Hu-Ya3] Hu, Pei-Chu; Yang, Chung-Chun A generalized $abc$-conjecture over function fields. J. Number Theory 94, No.2, 286-298 (2002).
  56. [Hu-Ya4] Hu, Pei-Chu; Yang, Chung-Chun Some progress in non-Archimedean analysis. Contemp. Math. 303, 37-50 (2002).
  57. [Ic] Ichimura, Humio A note on Greenberg's conjecture and the abc conjecture. Proc. Am. Math. Soc. 126, No.5, 1315-1320 (1998).
  58. [Ka] Katayama, Shin-ichi 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, 198-199 (1999).
  59. [Ki] Kim, Minhyong The ABC inequalities for some moduli spaces of log-general type. Math. Res. Lett. 5, no. 4, 517-522 (1998).
  60. [La1] Lang, Serge Old and new conjectured diophantine inequalities. Bull. Am. Math. Soc., New Ser. 23, No.1, 37-75 (1990).
  61. [La2] Lang, Serge Number theory III: Diophantine geometry. Encyclopaedia of Mathematical Sciences. 60. Berlin etc.: Springer-Verlag. xi, 296 p. (1991).
  62. [La3] Lang, Serge Die $abc$-Vermutung. (The $abc$-conjecture). Elem. Math. 48, No.3, 89-99 (1993).
  63. [Lan1] Langevin, Michel Partie sans facteur carre d'un produit d'entiers voisins. (Square-free divisor of a product of neighbouring integers). Approximations diophantiennes et nombres transcendants, C.-R. Colloq., Luminy/ Fr. 1990, 203-214 (1992).
  64. [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, 441-444 (1993).
  65. [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, 1031-1042 (1996).
  66. [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 17--22, 1996. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 19, 187-213 (1999).
  67. [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, 91-109 (1999).
  68. [Lo] Lockhart, P. On the discriminant of a hyperelliptic curve. Trans. Am. Math. Soc. 342, No.2, 729-852 (1994).
  69. [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, 207-222 (2003).
  70. [Ma] Mason, R.C. Diophantine Equations Over Function Fields. London Mathematical Society Lecture Note Series, 96. Cambridge etc.: Cambridge University Press. (1984).
  71. [Mas1] Masser, D.W. Note on a conjecture of Szpiro. Les pinceaux de courbes elliptiques, Semin., Paris/Fr. 1988, Asterisque 183, 19-23 (1990).
  72. [Mas2] Masser, D.W. On $abc$ and discriminants. Proc. Am. Math. Soc. 130, No.11, 3141-3150 (2002).
  73. [Mo] Moret-Bailly, 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, 37-58 (1990).
  74. [Mu] Mueller, Julia The $abc$-inequality and the generalized Fermat equation in function fields. Acta Arith. 64, No.1, 7-18 (1993).
  75. [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 3--6, 1996 on the occasion of the 10th anniversary of the Ramanujan Mathematical Society. Providence, RI: American Mathematical Society. Contemp. Math. 210, 85-95 (1998).
  76. [Mur-Wo] 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), 43--54, A K Peters, Natick, MA, 2002.
  77. [Ni1] Nitaj, Abderrahmane An algorithm for finding good $abc$-examples. C. R. Acad. Sci., Paris, Ser. I 317, No.9, 811-815 (1993).
  78. [Ni2] Nitaj, Abderrahmane Algorithms for finding good examples for the $abc$ and Szpiro conjectures. Exp. Math. 2, No.3, 223-230 (1993).
  79. [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 1993--94. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 235, 145-156 (1996).
  80. [Ni4] Nitaj, Abderrahmane La conjecture $abc$. (The $abc$ conjecture). Enseign. Math., II. Ser. 42, No.1-2, 3-24 (1996).
  81. [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, 165-186 (1988).
  82. [Ov] Overholt, Marius The diophantine equation $n!+1=m\sp 2$. Bull. Lond. Math. Soc. 25, No.2, 104 (1993).
  83. [Ri1] Ribenboim, Paulo $ABC$ candies. J. Number Theory 81, No.1, 48-60 (2000).
  84. [Ri2] Ribenboim, Paulo The (ABC) Conjecture and the radical index of integers. Acta Arith. 96, No.4, 389-404 (2001).
  85. [Ri3] Ribenboim, Paulo On square factors of terms of binary recurring sequences and the $ABC$ conjecture. Publ. Math. 59, No.3-4, 459-469 (2001). .
  86. [Ri-Wa] Ribenboim, Paulo; Walsh, Gary The ABC conjecture and the powerful part of terms in binary recurring sequences. J. Number Theory 74, No.1, 134-147, (1999).
  87. [Sc] Scanlon, Thomas The $abc$ theorem for commutative algebraic groups in characteristic $p$. Int. Math. Res. Not. 1997, No.18, 881-898 (1997).
  88. [Si] Silverman, Joseph H. Wieferich's criterion and the abc-conjecture. J. Number Theory 30, No.2, 226-237 (1988).
  89. [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), 60-100.
  90. [Sch] Schmidt, Wolfgang M. Diophantine approximations and diophantine equations. Lecture Notes in Mathematics. 1467. Berlin etc.: Springer-Verlag. viii, 217 p. (1991).
  91. [Sh] Shapiro, Harold N.; Sparer, Gerson H. Extension of a theorem of Mason. Commun. Pure Appl. Math. 47, No.5, 711-718 (1994).
  92. [Sm] Smirnov, A.L. Hurwitz inequalities for number fields. St. Petersbg. Math. J. 4, No.2, 357-375 (1993); translation from Algebra Anal. 4, No.2, 186-209 (1992).
  93. [Sny] Snyder, Noah An alternate proof of Mason's theorem. Elem. Math. 55, No.3, 93-94 (2000).
  94. [Ste-Ti] Stewart, C.L.; Tijdeman, R. On the Oesterle-Masser conjecture. Monatsh. Math. 102, 251-257 (1986).
  95. [Ste-Yu1] Stewart, C.L.; Yu, Kunrui On the $abc$ conjecture. Math. Ann. 291, No.2, 225-230 (1991).
  96. [Ste-Yu2] Stewart, C.L.; Yu, Kunrui On the $abc$ conjecture. II Duke Math. J. 108 No.1 169-181 (2001).
  97. [Sto] Stothers, W.W. Polynomial identities and Hauptmoduln. [J] Q. J. Math., Oxf. II. Ser. 32, 349-370 (1981).
  98. [Ti1] Tijdeman, R. Diophantine equations and diophantine approximations. Number theory and applications, Proc. NATO ASI, Banff/Can. 1988, NATO ASI Ser., Ser. C 265, 215-243 (1989).
  99. [Ti2] Tijdeman, R. Riemann's Hypothesis and the ABC-conjecture. Summer course 1999: unproven conjectures. Amsterdam: Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica. CWI Syllabus. 45, 31-44 (1999).
  100. [Va] Vaserstein, L. Quantum (abc)-theorems. J. Number Theory 81, No.2 , 351-368 (2000).
  101. [Vo1] Vojta, Paul Diophantine Approximations and Value Distribution Theory. Lecture Notes in Mathematics, 1239. Berlin etc.: Springer-Verlag. X, 132 p. (1987).
  102. [Vo2] Vojta, Paul A more general $abc$ conjecture. Int. Math. Res. Not. 1998, No.21, 1103-1116 (1998).
  103. [Vo3] Vojta, Paul On the $abc$ conjecture and diophantine approximation by rational points. Am. J. Math. 122, No.4, 843-872 (2000).
  104. [Vo4] Vojta, Paul Correction to ``On the $abc$ conjecture and diophantine approximation by rational points''. Am. J. Math. 123, No.2, 383-384 (2001).
  105. [Vu-Do] Vu Hoai An; Doan Quang Manh The ``abc'' conjecture for $p$-adic functions of several variables. Southeast Asian Bull. Math. 27, No.6, 959-972 (2004).
  106. [Wa1] Walsh, P.G. On integer solutions to $x\sp 2-dy\sp 2=1, z\sp 2-2dy\sp 2=1$. Acta Arith. 82, No.1, 69-76 (1997).
  107. [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 30--July 9, 1997. Volume 1: Diophantine problems and polynomials. Berlin: de Gruyter. 577-582 (1999).
  108. [Wan] Wang, Julie Tzu-Yueh A note on Wronskians and the ABC theorem in function fields of prime characteristic. Manuscr. Math. 98, No.2, 255-264 (1999).
  109. [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).
  110. [We2] de Weger, Benjamin M.M. $A+B=C$ and big Sha's. Q. J. Math., Oxf. II. Ser. 49, No.193, 105-128 (1998).
  111. [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, 107-137 (1995); Correction 74, No.4, 387 (1996).
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