The abc conjecture is one of the most famous unsolved problems in number theory. The conjecture claims for each real $ε> 0$ that there are only a finite number of coprime positive integer solutions to the equation $a+b = c$ with $c > (rad(a b c))^{1+ε}$. If true, the abc conjecture would imply many other famous theorems and conjectures as corollaries. In this paper, we discuss the abc conjecture and find new applications to powerful numbers, which are integers $n$ for which $p^2 | n$ for every prime $p$ such that $p | n$. We answer several questions from an earlier paper on this topic, assuming the truth of the abc conjecture.
The ABC conjecture of Masser and Oesterle' states that if (a,b,c) are coprime integers with a + b + c = 0, then sup(|a|,|b|,|c|) < c_e (rad(abc))^{1+e} for any e > 0. Oesterle' has observed that if the ABC conjecture holds for all (a,b,c) with 16 | abc, then the full ABC conjecture holds. We extend that result to show that, for every integer N, the "congruence ABC conjecture" that ABC holds for all (a,b,c) with N|abc implies the full ABC conjecture.
Ellenberg proved that the abc conjecture would follow if this conjecture were known for sums $a+b=c$ such that $D\mid abc$ for some integer~$D$. Mochizuki proved a theorem with an opposite restriction, that the full abc conjecture would follow if it were known for abc sums that are not highly divisible. We prove both theorems for general number fields.
The conjecture of Masser-Oesterlé, popularly known as $abc$-conjecture have many consequences. We use an explicit version due to Baker to solve a number of conjectures.
We prove that the $abc$-Conjecture implies upper bounds on Zsigmondy sets that are uniform over families of unicritical polynomials over number fields. As an application, we use the $abc$-Conjecture to prove that there exist uniform bounds on the index of the associated arboreal Galois representations.
This note imparts heuristic arguments and theorectical evidences that contradict the abc conjecture over the rational numbers. In addition, the rudimentary datails for transforming this problem into the doimain of equidistribution theory are provided.
In this paper, we define $X$-base Fibonacci-Wieferich prime which is a generalized Wieferich prime where $X$ is a finite set of algebraic numbers. We are going to show that there are infinitely many non-$X$-base Fibonacci-Wieferich primes assuming the $abc$-conjecture of Masser-Oesterlé-Szpiro for number fields. We also provide a new conjecture concerning the rank of free part of abelian group generated by all elements in $X$, and we will use the arithmetic point of view and geometric point of view to give heuristic.
This note formulates a conjecture generalizing both the abc conjecture of Masser-Oesterlé and the author's diophantine conjecture for algebraic points of bounded degree. It also shows that the new conjecture is implied by the earlier conjecture. As with most of the author's conjectures, this new conjecture stems from analogies with Nevanlinna theory; in this case it corresponds to a Second Main Theorem in Nevanlinna theory with truncated counting functions. The original abc conjecture of Masser and Oesterlé corresponds to the Second Main Theorem with truncated counting functions on P^1 for the…
We show that the $abc$ conjecture of Masser-Oesterlé-Szpiro for number fields implies that there are infinitely many non-Fibonacci-Wieferich primes. We also provide a new heuristic for the number of such primes beneath a certain value.
An extended generating series of the radical of n, involving two variables, leads to an identity in said variables, which proves Bombieri's abc-Conjecture for certain sets of integers.
Enrico Bombieri proved that the ABC Conjecture implies Roth's theorem in 1994. This paper concerns the other direction. In making use of Bombieri's and Van der Poorten's explicit formula for the coefficients of the regular continued fractions of algebraic numbers, we prove that Roth's theorem implies a weakened non-effective version of the ABC Conjecture in certain cases relating to roots.
In this short note we show that the uniform abc-conjecture over number fields puts strong restrictions on the coordinates of rational points on elliptic curves. For the proof we use a variant of the uniform abc-conjecture over number fields formulated by Mochizuki. As an application, we generalize a result of Silverman on elliptic non-Wieferich primes.
This note outlines a constructive proof of a proposition in Mochizuki's paper "Arithmetic elliptic curves in general position," making a direct use of computable non-critical Belyi maps to effectively reduce the full $abc$-conjecture to a restricted form. Such a reduction means that an effective $abc$-theorem is implied by Theorem 1.10 of Mochizuki's final IUT paper ("Inter-universal Teichmuller theory IV: log-volume computations and set-theoretic foundations").
In this short note we confirm the relation between the generalized $abc$-conjecture and the $p$-rationality of number fields. Namely, we prove that given K$/\mathbb{Q}$ a real quadratic extension or an imaginary $S_3$-extension, if the generalized $abc$-conjecture holds in K, then there exist at least $c\,\log X$ prime numbers $p \leq X$ for which K is $p$-rational, here $c$ is some nonzero constant depending on K. The real quadratic case was recently suggested by Böckle-Guiraud-Kalyanswamy-Khare.
Let $K/\mathbb{Q}$ be an algebraic number field of class number one and $\mathcal{O}_K$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $\mathcal{O}_K$ under the assumption of the \textit{abc} conjecture for number fields.
We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer $k\geq 2$ there are $\gg \log x$ Lucas non-Wieferich primes $p\leq x$ such that $p\equiv\pm1\pmod{k}$, assuming the $abc$ conjecture for number fields. Further, we discuss some applications of Lucas sequences in Cryptography.
Following N. Elkies ("ABC implies Mordell") we show that the abc conjecture of Masser-Oesterle implies an effective version of Siegel's theorem about integral points on algebraic curves, i.e. an upper bound for the S-integral points where the dependence on S is explicit. The converse statement is also announced in this note. For both results, the main geometric tool is a theorem of G.V. Belyi.
We prove that for any positive integer c there are at least N(c), $1\leq N(c) < ϕ(c)/2$ representations of c as a sum of two positive integers a, b, with no common divisor, such that the N(c) radicals R(abc) are all greater than kc, where k an absolute constant.
We prove that for a positive integer $c$ and any given $\varepsilon$, $0<\varepsilon<1$, the number $N(c)$ of equations $c=a+b$, $a<b$, with positive coprime integers $a$ and $b$, which satisfy the inequality $$c < R(c)^{\frac{\varepsilon}{1+\varepsilon}}R(a)^{\frac{1}{1+\varepsilon}}R(b)^{\frac{1}{1+\varepsilon}},$$ where R(n) is the radical of $n$, is for $c\to\infty$ $$N(c)=(1-\varepsilon)\frac{ϕ(c)}{2}+O\Bigl(\frac{ϕ(c)}{2}\Bigr).$$ An analogue for the abc-conjecture inequality $c<R(abc)^{1+\varepsilon}$ (without a constant factor) will also be proved.
Assuming the abc conjecture with $ε=1/6$, we use elementary methods to show that only finitely many $s$-Cullen numbers are repunits, aside from two known infinite families. More precisely, only finitely many positive integers $s$, $n$, $b$, and $q$ with $s,b \geq 2$ and $n,q \geq 3$ satisfy \[C_{s,n} = ns^n + 1 = \frac{b^q -1}{b-1}.\]
We consider a variant of the ABC Conjecture, attempting to count the number of solutions to $A+B+C=0$, in relatively prime integers $A,B,C$ each of absolute value less than $N$ with $r(A)<|A|^a, r(B)<|B|^b, r(C)<|C|^c.$ The ABC Conjecture is equivalent to the statement that for $a+b+c<1$, the number of solutions is bounded independently of $N$. If $a+b+c \geq 1$, it is conjectured that the number of solutions is asymptotically $N^{a+b+c-1 \pm ε}.$ We prove this conjecture as long as $a+b+c \geq 2.$
We address the Uniform Boundedness Conjecture of Morton and Silverman in the case of unicritical polynomials, assuming a generalization of the $abc$-conjecture. For unicritical polynomials of degree at least five, we require only the standard $abc$-conjecture.
The abc conjecture, one of the most famous open problems in number theory, claims that three positive integers satisfying a+b=c cannot simultaneously have significant repetition among their prime factors; in particular, the product of the distinct primes dividing the three integers should never be much less than c. Triples of numbers satisfying a+b=c are called abc triples if the product of their distinct prime divisors is strictly less than c. We catalog what is known about abc triples, both numerical examples found through computation and infinite familes of examples established theoreticall…
We prove that for any positive integer c and any s > 0 there are representations of c as a sum a+b of two coprime positive integers a, b, such that the respective radicals are all greater than K(s)R(c)^(1-s)c^2. For the reprasentations in question, this is a stronger result than the abc-conjecture, which postulates that these representations are greater than k(s)c^1/(1+s).
We show that Vojta's conjecture for some rational surfaces is related to the $abc$ conjecture. More specifically, we prove that Vojta's conjecture on these surfaces implies a special case of the $abc$ conjecture, while the $abc$ conjecture implies Vojta's conjecture on these surfaces. Moreover, for similar but different rational surfaces, we prove Vojta's conjecture unconditionally. To prove these results, we use some (possibly new) properties of Farey sequences.
This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor of ABC. We consider primitive solutions (gcd(A, B, C)=1) having smoothness no larger than a fixed power p of log H. Assuming the abc Conjecture we show that there are finitely many solutions if p<1. We discuss a conditional result, showing that the Generalized Riemann Hypothesis (GRH) implies there are infinitely many primitive solutions when p>8. We sketch some details of the proof o…
Nous montrons qu'un raffinement du théorème de Siegel sur les points entiers de courbes algébriques impliquerait la conjecture abc de Masser-Oesterlé. Nous formulons une hypothèse "Siegel uniforme" qui est une majoration de la hauteur des points S-entiers de la courbe, en termes du corps de rationalité et de l'ensemble de places S. La validité de l'hypothèse pour une quelconque courbe algébrique de caractéristique d'Euler-Poincaré strictement négative, impliquerait une version de la conjecture abc. Ceci étend aux points S-entiers des résultats précédents de L. Moret-Bailly, et est en quelque s…
This note studies various ways of measuring the complexity of primitive solutions to the equation $A \,+\, B\, + \,C = 0$ and conjectures relating them. We define the insulator $\mathcal I(A,B,C)$ of a primitive solution as the smallest positive integer $\mathcal I$ such that the primes dividing the product $ABC \cdot \mathcal I$ are exactly those below a given bound. We show that the strong XYZ Conjecture of Lagarias and Soundararajan implies there are only finitely many primitive solutions $(A,B,C)$ with a given insulator.
Our main source of inspiration was a talk by Hendrik Lenstra on harmonic numbers, which are numbers whose only prime factors are two or three. Gersonides proved 675 years ago that one can be written as a difference of harmonic numbers in only four ways: 2-1, 3-2, 4-3, and 9-8. We investigate which numbers other than one can or cannot be written as a difference of harmonic numbers and we look at their connection to the $abc$-conjecture. We find that there are only eleven numbers less than 100 that cannot be written as a difference of harmonic numbers (we call these $ndh$-numbers). The smallest …
We establish a precise correspondence between the ABC Conjecture and N=4 super-Yang-Mills theory. This is achieved by combining three ingredients: (i) Elkies' method of mapping ABC-triples to elliptic curves in his demonstration that ABC implies Mordell/Faltings; (ii) an explicit pair of elliptic curve and associated Belyi map given by Khadjavi-Scharaschkin; and (iii) the fact that the bipartite brane-tiling/dimer model for a gauge theory with toric moduli space is a particular dessin d'enfant in the sense of Grothendieck. We explore this correspondence for the highest quality ABC-triples as w…
The $abc$ conjecture predicts a highly non trivial upper bound for the height of an algebraic point in terms of its discriminant and its intersection with a fixed divisor of the projective line counted without multiplicity. We describe the two independent proofs of the strong $abc$ conjecture over function fields given by McQuillan and Yamanoi. The first proof relies on tools from differential and algebraic geometry; the second relies on analytic and topological methods. They correspond respectively to the Nevanlinna and the Ahlfors approach to the Nevanlinna Second Main Theorem.
We develop a general framework to study Szpiro's conjecture and the $abc$ conjecture by means of Shimura curves and their maps to elliptic curves, introducing new techniques that allow us to obtain several unconditional results for these conjectures. We first prove various general results about modular and Shimura curves, including bounds for the Manin constant in the case of additive reduction, a detailed study of maps from Shimura curves to elliptic curves and comparisons between their degrees, and lower bounds for the Petersson norm of integral modular forms on Shimura curves. Our main appl…
While currently the $abc$ conjecture and work towards it remains open or is disputed, at the same time much work has been done on weaker versions, as well as on its generalisation to number fields. Given integers satisfying $a+b=c$, Stewart and Yu were able to give an exponential bound for $\max(a,\,b,\,c)$ in terms of the radical over the integers, while Györy was able to give an exponential bound in the algebraic number field case for the projective height $H_{K}(a,\,b,\,c)$ in terms of the radical for algebraic numbers. We generalise Stewart and Yu's method to give an improvement on Györy's…
For an odd positive integer $n\ge 5$, assuming the truth of the $abc$ conjecture, we show that for a positive proportion of pairs $(a,b)$ of integers the trinomials of the form $t^n+at+b (a,b\in \mathbb Z)$ are irreducible and their discriminants are squarefree.
We show that an earlier conjecture of the author, on diophantine approximation of rational points on varieties, implies the ``abc conjecture'' of Masser and Oesterl'e. In fact, a weak form of the former conjecture is sufficient, involving an extra hypothesis that the variety and divisor admit a faithful group action of a certain type. Analogues of this weaker conjecture are proved in the split function field case of characteristic zero, and in the case of holomorphic curves (Nevanlinna theory). The proof of the latter involves a geometric generalization of the classical lemma on the logarithmi…
We show a weak form of the function field version of Oesterle's abc conjecture. It asserts that, if $B$ is a complex projective connected curve, the number of intersection points, counted without multiplicities, of a fixed divisor $D$ of degree $d>0$ over $B$ with the graph $H$ of a section $h:B\to B\times \bP^1$ to the first projection is at least $(d-2)n-C(B,D)$, where $n$ is the degree of $H$ over $\bP^1$, and $C(D,B)$ a constant depending only on these two data. We show this number is at least $(d-2[\sqrt {d}]).n-C(D,B)$. The constant is ineffective.
The Weakly Special Conjecture contradicts orbifold Mordell, and hence the abc conjecture
Radboud University（拉德堡德大学）；Université de Lorraine（洛林大学）；Institut Élie Cartan de Lorraine, Université de Lorraine（洛林大学埃利·嘉当研究所）；IMAPP Radboud University Nijmegen（奈梅亨拉德堡德大学IMAPP）；Laboratoire Analyse, Géométrie et Applications, Institut Galilée - Université Sorbonne Paris Nord（索邦巴黎北大学吉莱研究所分析与几何应用实验室）
We define certain arithmetic derivatives on $\mathbb{Z}$ that respect the Leibniz rule, are additive for a chosen equation $a+b=c$, and satisfy a suitable non-degeneracy condition. Using Geometry of Numbers, we unconditionally show their existence with controlled size. We prove that any power-saving improvement on our size bounds would give a version of the $abc$ Conjecture. In fact, we show that the existence of sufficiently small arithmetic derivatives in our sense is equivalent to the $abc$ Conjecture. Our results give an explicit manifestation of an analogy suggested by Vojta in the eighti…
This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption and under assumption of the abc-conjecture. Finally we prove that the explicit abc-conjecture implies the Erdős-Woods conjecture for each k>2.
