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 January 16, 2023

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

Comment by Preda Mihailescu, October 13, 2022
"Part of the community is skeptical about the results in the paper of Shinichi Mochizuki, due to a published criticism of Peter Scholze and Jakob Stix, to which Mochizuki answered, but the answer was not accepted by the other side. The paper was however reviewed by an international commitee of 12, and this after the mentioned comments. Since the reviewers accepted publication, it is very likely that they found their own way to convince that the criticism did not point to an actual error of the work. We may expect that in the near future, some members of the review commitee will provide a more detailed answer to the criticism, and engage in a fruitful debate on the subject, to the benefit of the entire community.
The Inter-universal Teichmüller (IUT papers were published in a special volume of Publications of the Research Institute for Mathematical Sciences (RIMS),, whose specially appointed 14 editors include A. Tamagawa, the top world expert in anabelian geometry, M. Kashiwara, the Kyoto Prize and Chern Medal winner, T. Mochizuki (Takura Mochizuki, not related to S. Mochizuki), the Breakthrough Prize winner, H. Nakajima, the current President of the IMU. There were ten revisions of the IUT papers before their publication (
Here are more papers of S. Mochizuki).
The editors were well aware of a 10 pages 2018 report by P. Scholze and P. Stix based on an own “model version of IUT” – on this base, they claimed having found an error, in the last hours of their visit to Japan. They received a detailed answer from S. Mochizuki, but the communication stopped there. The investigation done by the reviewers and editors obviously led to the conclusion that the reply is consistent and there is no error, the issue being probably related with discrepancy between the actual, complex theory, and the model. It is a fact that there are precious few outstanding international experts in anabelian geometry, that can hope to understand the IUT – and none of those outside the reviewers and several more, had the material time necessary for studying in depth this new theory. All other members of the community are reduced to relying on second and third hand “opinions” and “conclusions”. Personally, I am also one of those trying to build an own insight by confronting the points of view of various sides. It must be added that the original IUT did not provide an effective proof of ABC; this was first achieved in 2020 by a group of authors, including Mochizuki, and appeared in
Kodai Math. J. 45(2): 175-236 (June 2022), see also the fine introduction by I. Fesenko
Given the great importance of these results for the development of mathematics, it is to be hoped that the interrupted dialogue may be resumed, with the help of the editorial and reviewing committee on one side, and the two authors of the report and possible further experts with at least comparable insight in IUT, on the other, as well as the desire for clarification expressed by numerous members of community. Such a broad exchange should help reach a common clarity about the critical points, much to the profit of the community at large. Otherwise, it is both hard to conceive for a common mathematician, that such a competent editorial board would misunderstand a valid criticism and at the same time it is hard to rely upon a result which was criticized by well established mathematicians, albeit without reciprocal clarification with the first group. Knock knock at the door, it is said!"


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.

abc conjecture: Given any  ε > 0, there exists a constant C(ε) > 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+ε .

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 :
For any ε > 0, there exists a constant C(ε) > 0 such that for every triple a,b,c of pairwise coprime positive integers satisfying a + b = c, we have

c < rad(abc)1+ε .

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


  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

    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.
    Let F(X,Y) be a homogenous polynomial with integer coefficients and no repeated linear factors. For any ε > 0, there exists a constant Cε,F such that for any coprime integers m and n, 

    max(|m|,|n|)   Cε,F (rad(mnF(m,n)))deg(F)+ε.

    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.
    Given any integer n > 2 and any ε> 0, there exists a constant Cn,ε 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,ε(rad(a1 × ... × an))2n-5+ε.

  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.
    Given any ε> 0, there exists a constant Cε> 0 such that for every triple of positive integers a,b, c, satisfying a+b=c and gcd(a,b)=1 we have

     Cε- rad(abc))1+ε,

    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

    K(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),

    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 ε >0, there exists a constant C(k,ε) such that

    max|ai| < C(k,ε)R(a0...ak)1+ε,


    R(a0...ak) =  i rk-1(ai).

    If k=2, this corresponds to the abc conjecture.

  9. A new abc-related conjecture for integers. (Suggested by Dao Thanh Oai, January 11, 2023, ). Let P be a positive integer with factorization 

    P=p1a1 × p2a2 × p3a3× ....× pkak.

    Define h(P) by h(1)=1, and h(P)=min(a1,a2,...,ak)
    New conjecture: If P1,P2,...,Pn are positive pairwaise prime
    integers, then

    h(Pn),h(P1+P2+...+Pn))≤ n+1.

    For n=2, the new conjecture asserts that min(h(P1), h(P2),h(P1+P2))≤ 3. This has been checked by Dao Thanh Oai up to 1018.
    The new conjecture implies Euler's sum of powers conjecture:  if a1, a2, a3 .....,an, an+1 are positive integers such that a1k+a2k...+ank =an+1k, then k≤ n. 
    For n=2, there are infinitely many positive coprime integers P1, P2 such that min(h(P1), h(P2),h(P1+P2))= 3 (see [Ni5]).       

 To Index


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.

    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.
                              Given a positive integer a> 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]:
    Given any ε> 0, there exists a constant Cε> 0 such that for every positive integers x and y such that x3 - y2 is non-zero then

    |x3 - y2|> Cε max(x3, y2)1/6-ε.

  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 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].

  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 ε> 0, there exists a constant Cε> 0 such that for every elliptic curve with minimal discriminant   and conductor N, we have | | < CεN6+ε.
    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 Cε, 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+ε, with ε > 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 ε> 0 there is a constant Cε such that if a< b are positive integers having the same prime factors, then

    b-a> Cε a1/2-ε.

  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 ax
    2+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.

    Let F(X) be a polynomial with integer coefficients and no repeated roots. For any ε > 0, there exists a constant Cε,F such that for any integer n,

    |n|deg(F)-1-ε  Cε,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 ε > 0, there exists a positive constant Cε 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) > U} and {n > 0, w(Vn) > V} 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. 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. 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). 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

     sz = ____________________.

    and FE/K is the conductor of E/K.

  26. Lang's Integral Point Conjecture (1978). 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.  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.  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),

    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 weakened form of the abc conjecture.

  29. The diophantine equation pv-pw=qx-qy 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.  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.

  31. The ideal Waring’s Theorem.  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.

  32. Uniform bounds on dynamical Zsigmondy sets. In [Loo], N. Looper proved that the abc conjecture for number fields implies that there are explicit upper bounds for on Zsigmondy sets that are uniform over families of unicritical polynomials over number fields.

  33. Roth's Theorem implies a Weakened Version of the ABC Conjecture for Special Cases. In [Sib], P. Sibbertsen, T. Lampert, K. Müller, and M. Taktikos introduced the weakened non-effective ABC Conjecture as follows.
    For every  ε> 0, there is a constant Kε such that for all triples (a,b,c) of coprime positive integers et all positive integers m,n,k with am+bm=ck, we have

    ck< Kε (abc)1+ε.

    They proved that Roth’s theorem implies this version of the abc Conjecture in certain cases relating to roots.

 To Index


For any triple of positive integers a, b, c satisfying a+b=c and gcd(a,b)=1 let



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


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 










310 × 109






32 × 56 × 73

221 × 23




19 × 1307

7 × 292 × 318

28 × 322 × 54





511 × 132

28 × 38 × 173


J.B-J.B, A.N 



2 × 37

54 × 7






211 × 29

1. 54708



72 × 412 × 3113

1116 × 132 × 79

2 × 33 × 523 × 953





29 × 317 × 132

115 × 17 × 313 × 137




13 × 196

230 × 5

313 × 112 × 31




318 × 23 × 2269

173 × 29 × 318

210 × 52 × 715



de Smit's HTML complete list of good abc-examples. 

[PDF]   Complete list of all known good abc-examples. 

[PDF]  T. Dokchitser list of new good abc-examples. 

Table II. The top ten good abc-Szpiro-examples









13 × 196

230 × 5

313 × 112 × 31

4. 41901



25 × 112 × 19

515 × 372 × 47 

37 × 711 × 743 




219 × 13 × 103


311 × 53 × 112




198 × 434 × 1492

215 × 523 × 101

313 × 13× 292 × 376 × 911




235 × 72 × 172 × 19 

327 × 107

515 × 372 × 2311 




318 × 23 × 2269 

173 × 29 × 318

210 × 52 × 715 




174 × 793 × 211 

229 × 23 × 292





514 × 19 

25 × 3 × 713

117 × 372 × 353 




27 × 54 × 722

194 × 37× 474 × 536

314 × 11× 139 × 191 × 7829





72 × 116 × 199

2 × 138 × 17



[PDF] Complete list of all known good abc-Szpiro examples. 

[PDF]  T. Dokchitser's list of new good abc-Szpiro examples. 

Table III. The top ten good purely algebraic 
abc-examples over K=Q(d)


















wi3-2wi2+4wi-4=0, i=1,2,3




de Weger's example 




wi3+3wi-1=0, i=1,2,3




Dokchitser's example 








Dokchitser's example 
























Dokchitser's example 















3 × 76






















24× 32×(1-2w)


















132× (1+w)7w3






(8-3w)2 (5-2w)

(8-3w)7 (3-w)3 (5+2w)12







72(5+2w)9 (2-w)9 (3-w) (1+w) (1-w)






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 

Latest authors of good abc-examples:
F.R. : Frank Rubin
I.C. : Ismael Jimenez Calvo






March 2, 2019

25× 678× 107×22381

58 ×536 ×3535

322 × 714× 43 × 83



May 15, 2017

313× 375× 449392

55 ×723 ×19×463×863

220 × 538 × 61×1134



December 4, 2016

33× 2413

58 ×119 ×19×613

215 × 172 × 331×10614



August 1, 2014

230× 54× 47

35 ×175 ×293×5413

7 × 112 × 13 × 2311 × 127



August 8, 2014


5 ×135 ×194 ×232 ×2332 ×301591

235 × 310 × 117 ×53



June 6, 2013

436× 672×29209

318 ×79 ×29×373

29 × 58 × 177 ×234



May 25, 2013

25× 675× 2632×487

231 ×17×236 ×29

316 × 515 × 112



November 11, 2010


32 ×1318 ×613

223 × 174 × 2512 ×17333



August 8, 2010

220× 2335

37 ×598 ×47292

324 × 56 ×19× 23 × 2512



June 25, 2010

221× 317× 31

52× 418× 170532

119 × 61  × 712 × 4313



June 12, 2010

7× 167 × 8114× 919

34 ×132 ×2312×674

231 × 53 × 112×175×1074



May 22, 2010

25× 55× 75×113×292×3472

38 ×978 ×10912

1312 × 197 × 2939



April 23, 2010


213 ×139 ×29×2441×76732

319 ×114 ×4635



March 05, 2010

11×198×23× 675×1877





February 25, 2010

3117× 6029

245 ×56 ×8392

38 ×73 ×176 ×432 ×1573



February 12, 2010

237× 312×21093

513 ×1315 ×2939

723 × 11× 793345871



January 28, 2010

214× 36× 424873

514 ×2912 ×83

78 × 113 × 477 × 4610911



Table V. Largest good 
abc-examples, sorted by number of digits


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

Number of digits of c






214× 36× 424873

514 ×2912 ×83

78 × 113 × 477 × 4610911




237× 312×21093

513 ×1315 ×2939

723 × 11× 793345871




238 × 374

228 × 37 ×114 ×193 × 61 ×127×1732

518 × 174 × 432× 48172




224 × 55× 475 × 1812

1314 × 19× 103 × 5712× 4261

728 × 17× 372




59 × 172× 234 × 372× 43× 4817

314 × 118 × 612 × 1734

252 × 196 × 1272




312× 617× 3889

323 ×117 × 1513 ×173

223 × 56 × 73× 833× 3493



 To Index


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abc Theses

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 e-mail address: abderrahmane.nitaj(at)
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