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 June 30, 2024

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

• Generalizations 

• Consequences

• Tables

  • The top ten good abc examples

  • The top ten good abc-Szpiro examples 

  • The top ten good algebraic abc examples 

  • New good abc examples 

  • Hunters of abc examples rekenmeemetabc, abc@home 

  • Largest good abc examples 

• Bibliography

• Links to abc papers

• abc Theses

• Contact

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  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.

• 1986, C.L. Stewart and R. Tijdeman [Ste-Ti]: c < exp(C1 rad(abc)¹⁵) where C₁ is an absolute constant.

• 1991, C.L. Stewart and Kunrui Yu [Ste-Yu1]: c < exp(C2 rad(abc)^{2/3+ ε} ) where  C₂ is a positive effectivley computable constant in terms of  ε.

• 2007, K. Gyory new results on the abc 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|)   C_ε,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 C_n,ε such that for all integers a₁, ..., a_n with a₁+...+ a_n=0, gcd( a₁,..., a_n)=1 and no proper zero subsum, we have

   max(|a₁|,...,|a_n|)   C_n,ε(rad(a₁ × ... × a_n))^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 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   C_ε(ε^- 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 V_K denote the set of primes on K, that is, any v in V_K is an equivalence class of non-trivial norms on K (finite or infinite). Let ||x||_v=N_K/Q(P)^-v_P^(x) if v is a prime definied by a prime ideal P of the ring of integers O_K in K and v_P is the corresponding valuation, where N_K/Q is the absolute norm. Let ||x||_v=|g(x)|^e for all 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

   
   H_K(a,b,c) =  _{v in VK} max(||a||_v, ||b||_v, ||c||_v), 

   and the radical (or conductor) of (a,b,c) by

   rad_K(a,b,c) =   _{P in IK(a,b,c)}N_K/Q(P),

   where I_K(a,b,c) is the set of all prime ideals P of O_K for which ||a||_v, ||b||_v , ||c||_v are not equal. Let D_K/Q denote the discriminant of K.

   • The abc conjecture for algebraic number fields.For any ε> 0, there exists a positive constant C_K,ε such that for all a,b,c in K^* satisfying a+b+c=0, we have

     H_K(a,b,c) < C_K,ε(rad_K(a,b,c))^1+ε.

   • The uniform abc conjecture.For any ε> 0, there exists a positive constant C_ε such that for all a,b,c in K^* satisfying a+b+c=0, we have

     H_K(a,b,c) < C_ε^[K:Q](|D_K/Q|rad_K(a,b,c))^1+ε.

   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 f_j(z), j=0...k, be k entire functions in K without common zeros, not all constants and no proper subsum is equal to 0 satisfying f₀+f₁+....+f_k = 0. In 2002, Hu and Yang [Hu-Ya3] showed that

   max{T(r,f_j)} < N(r,1/f₀, 1/f₁,..., 1/f_k))-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...p_nⁱ_n where p₁,...,p_n are distinct primes. Define the k-radical of a to be

   r_k(a)= _pj|a p_j^min(i_j^,k).

   In 2002, Hu and Yang [Hu-Ya3] proposed the following conjecture.
   Let a_i, i=0...k,  be nonzero integers without common factor and no proper subsum is equal to 0 such that

   a₀+.....+a_k =0.

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

   max|a_i| < C(k,ε)R(a₀...a_k)^1+ε,

   where

   R(a₀...a_k) =  _i r_k-1(a_i).

   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=p₁^a1 × p₂^a2 × p₃^a3× ....× p_k^ak.
   
   

   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
   

   
   min(h(P1), h(P2),...,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 a_1, a_2, a₃ .....,a_n, a_{n+1 are positive integers such that} a₁^k+a₂^k...+a_n^k =a_n+1^k, 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 Ax^r+By^s=Cz^t has only finitely many solutions in integers x, y, z, r, s, t satisfying gcd(x,y,z)=1 and 1/r+1/s+1/t< 1 . Note that Darmon and Granville [Dar-Gr] proved that if r, s, t are fixed with 1/r+1/s+1/t< 1 then the equation Ax^r+By^s=Cz^t has only finitely many solutions in pairwise coprime integers x,y,z.

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=y^m [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 C_ε> 0 such that for every elliptic curve with minimal discriminant   and conductor N, we have | | < C_εN^6+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/q^2+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/q^2+k for all but a finite number of fractions p/q in lowest form, where k= C(log q)^-1/2(log(log q))^-1 for some constant C depending only on  . (See also [Fr3]).

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 a^1/2-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))d^1/2(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.

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

    
    |n|^deg(F)-1-ε  C_ε,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(c_fMN) pairs of positive integers m < M, n < N for which f(m,n)/B' is squarefree as M, N tend to infinity, where c_f is a constant which depends only on f. A similar result exists for a polynomial f(x) with integer coefficients and no repeated roots.

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 | |  N^{1/2 +}  . 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 U_n=PU_n-1-QU_n-2 and V_n=PV_n-1-QV_n-2. Ribenboim and Walsh [Ri-Wa] proved that the abc conjecture implies that the sets {n > 0, w(U_n) > U_nε} and {n > 0, w(V_n) > V_nε} are finite. It follows that U_n and V_n 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 Z_p-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(d^1/2) with 0< |d|< x whose order of the class group is divisible by g. It has been proved that N(x) is infinite, but no quantitative bound is known. R. Murty [Mur] showed that the abc conjecture enables us to count N(x), namely that for any   > 0, there exists a positive constant C  such that N(x) > C  x^k+  where k=1/g if d < 0 and k=1/2g if d > 0.

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 g_n = N₁ⁿ+N'₁ⁿ, h_n = (N₁ⁿ-N'₁ⁿ)/(M²+/-4)^1/2, N₂=h_2n+1+(h²_2n+1-1)^1/2 and N₃=g_2n+1/M+(g²_2n+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((h²_2n+1-1)^1/2), N₃ is the fundamental unit of the real quadratic field Q((g²_2n+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, (h²_2n+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 |N_K/QD_E/K| where D_E/K is the minimal discriminant of E/K, H(P) is the canonical height of P and N_K/Q is the norm. This conjecture of Lang follows from the ABC conjecture (see [H-Si]). To be more precise, Hindry and Silverman proved that H(P)>(20sz )^-8[K:Q]10^-1.1-4szlog |N_K/QD_E/K| where sz is the Szpiro ratio

     log|NK/QDE/K|
     sz = ____________________.
     log|NK/QFE/K|

    and F_E/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...p_nⁱ_n where p₁,...,p_n are distinct primes. Define the k'th radical of n to be

    n_k(n)= _pjⁱ_j||n p_j^UB(i_j^/k),

    
    where UB(x) is the smallest integer greater than or equal to x. In 2002, Broughan [Broug] proposed the following conjecture:
    Let a, b, c be positive integers without common factor such that a+b=c. There exists a positive constant C(k,m) such that

    c < C(k,m)n_k(abc)^1+1/m.

    
    This conjecture is a weakened form of the abc conjecture.

29. The diophantine equation p^v-p^w=q^x-q^y.  In 2003, Luca [Lu] showed that assuming the abc conjecture, the diophantine equation p^v-p^w=q^x-q^y has only finitely many positive integer solutions p, q, v,w,x,y where p and q are distinct prime numbers.

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 x^k with x >0. For example, according to Lagrange's Theorem g(2)=4 and according to Wieferich's Theorem g(3)=9 (see [Wal1, Wal2]). The ideal Waring’s Theorem is the 1853 conjecture that asserts that for any  k>1, g(k)=2^k+[(3/2)^k]-2.
    In a personnal comunication to M. Waldschmidt (see [Wal1, Wal2]), S. David proved that the ideal Waring’s Theorem is a consequence of the abc conjecture for sufficiently large k.

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 a^m+b^m=c^k, we have

    c^k< 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

 log(c)
  α=________________________,
 log(rad(abc)) 


 log(abc)
  ρ=_________________________.
 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))
  γ=________________________________________.
 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 

       log(c)
α=_____________________,
     log(rad(abc))

         log(abc)
ρ=__________________________.
       log(rad(abc))

            log(HK(a,b,c))
 γ=___________________________________.
       log|DK/Q|+log(radK(a,b,c)))

        log(c)
α=_____________________,
     log(rad(abc))

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

        log(c)
α=_____________________,
     log(rad(abc))
      

 To Index

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