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


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 (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.

To Index

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.
  3. The abc conjecture for binary forms. It is shown in [Lan2] that the abc conjecture implies the following conjecture.

  4. 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.
  5. The n-term abc conjecture for integers. In 1994, Browkin and Brzezinski [Br-Brz] proposed the following conjecture.

  6. 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+.
  7. 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.

  8. 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(-rad(abc))1+,
    where  denotes the number of distinct prime factors of abc.
  9. 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
  10. 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:
  11. 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.
  12. 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.
  13. 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.
To Index

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
  3. 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.
  4. 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.

  5. Given a positive integer a> 1. There exist infinitely many primes p such that p2 does not divide ap-1-1.
  6. 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.
  7. 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.
  8. Hall's conjecture. The abc conjecture implies the following weak form of the Hall conjecture [Ni4, Sch]:

  9. 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-.
  10. 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.
  11. Brown Numbers and Brocard's Problem. Pairs of numbers (m,n) satisfying Brocard'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].
  12. 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.
  13. 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]).
  14. 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).
  15. 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]).
  16. 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
  17. b-a> C a1/2-.
  18. 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".
  19. Power free-values of polynomials. Langevin noted in [Lan2] the following conjecture which is a consequence of the abc conjecture.

  20. 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)).
  21. 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.
  22. 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.
  23. 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.
  24. 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.
  25. 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.
  26. 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.
  27. 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.
  28. 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.
  29. 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.
  30. 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.
  31. 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.
  32. 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.
  33. 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.
  34. 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.
  35. 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.
  36. 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 xk 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)=2k+[(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

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 
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 × 19 515 × 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 × 107 515 × 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
I.C. : Ismael Jimenez Calvo
J.W. : Jarek Wroblewski

Date a b c Author
May 22, 2010 25× 55× 75×113×292×3472 38 ×978 ×10912 1312 × 197 × 2939 1.4128 F.R.
April 23, 2010 54×1913×103 213 ×139 ×29×2441×76732 319 ×114 ×4635 1.4494 F.R.
March 05, 2010 11×198×23× 675×1877 312×473×835×1133 217×522×10192 1.4019 F.R.
February 25, 2010 3117× 6029 245 ×56 ×8392 38 ×73 ×176 ×432 ×1573 1.4229 F.R.
February 12, 2010 237× 312×21093 513 ×1315 ×2939 723 × 11× 793345871 1.4121 F.R.
January 28, 2010 214× 36× 424873 514 ×2912 ×83 78 × 113 × 477 × 4610911 1.4126 F.R.
May 02, 2009 312× 617× 3889 323 ×117 × 1513 ×173 223 × 56 × 73× 833× 3493 1.4150 J.W.
April 22, 2009 75× 61 213 ×137 × 173 × 42293 313 × 58 × 113× 53× 732 ×892× 103 1.4077 J.W.
March 01, 2009 412× 592 38 ×76 ×138 ×1831 212 × 54 × 766513 1.4072 F.R.
January 02, 2009 35× 515× 135 710 ×795 × 35323 211 × 737 × 832× 197 1.4091 F.R.
December 25, 2008 212× 133× 2233 315 ×113 × 975 ×409 515 × 1794 × 2141 1.4123 F.R.
December 25, 2008 2× 510× 134 315 ×7× 317 × 45817 118 × 1092 × 36773 1.4232 F.R.
December 20, 2008 233 × 536× 31672 28 × 329 × 113992 57 × 74 × 1312× 523 1.4501 F.R.
May 09, 2008 238 × 374 228 × 37 ×114 ×193 × 61 ×127×1732 518 × 174 × 432× 48172 1.4502 I.C.
December 20, 2007 52 × 134 174 × 141971 318 × 74 × 113× 894 1.4226 F.R.
September 20, 2007 52 × 2310× 106531 711 × 113 × 1934 24 × 319 × 178× 29 1.4646 F.R.
September 07, 2007 224 × 55× 475 × 1812 1314 × 19× 103 × 5712× 4261 728 × 17× 372 1.447420 F.R.
Auguste 26, 2007 28 × 47× 16421 512 × 4396 259 × 41× 73939 1.4017 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.
Auguste 07, 2007 321 × 7× 4498001 510 × 4995 228 × 173 × 475 1.4051 F.R.
August 03, 2007 313 × 615 1710 × 832 × 41059619 2 × 33 × 517 × 712 1.451917 F.R.
August 03, 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 07, 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
30 214× 36× 424873 514 ×2912 ×83 78 × 113 × 477 × 4610911 1.4126 F.R.
30 237× 312×21093 513 ×1315 ×2939 723 × 11× 793345871 1.4121 F.R.
29 238 × 374 228 × 37 ×114 ×193 × 61 ×127×1732 518 × 174 × 432× 48172 1.4502 I.C.
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.
28 312× 617× 3889 323 ×117 × 1513 ×173 223 × 56 × 73× 833× 3493 1.4150 J.W.
To Index

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