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.
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.
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 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.
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.
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 \[C_{s,n} = ns^n + 1 = \frac{b^q -1}{b-1}.\]
Let K be a number field, let f(x) in K(x) be a rational function of degree d> 1, and let z in K be a wandering point such that f^n(z) is nonzero for all n > 0. We prove that if the abc-conjecture holds for K, then for all but finitely many positive integers n, there is a prime p of K such that p | f^n(z) and p does not divide f^m(z) for all positive integers m < n. We prove the same result unconditionally for function fields of characteristic 0 when f is not isotrivial.
Let $K$ be a number field or a function field of characteristic 0. If $K$ is a number field, assume the $abc$-conjecture for $K$. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in $K(x)$ that are not postcritically finite. For example, suppose $K$ is a number field and $f\in K[x]$ is not postcritically finite, and let $K_n$ be the field generated by the $n$th iterated preimages under $f$ of $β\in K$. We show that for all large $n$, there is a prime of $K$ that ramifies in $K_n$ and does not ramify in $K_m$ for any $m<n$.
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 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.
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").
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 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 a congruence discovered by George E. Andrews in 1969 for the Fibonacci quotient directly implies a simpler congruence found by Hugh C. Williams in 1991.
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.
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…
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…
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.
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…
Let p be any prime and a be a positive integer. For nonnegative integers l,n and an integer r, the normalized cyclotomic $ψ$-coefficient $${n,r}_{l,p^a}:=p^{-[(n-p^{a-1}-lp^a)/(p^{a-1}(p-1))]} \sum_{k=r(mod p^a)}(-1)^k{n \choose k}{{(k-r)/p^a} \choose l}$$ is known to be an integer. In this paper, we show that this coefficient behaves like binomial coefficients and satisfies some Lucas-type congruences. This implies that a congruence of Wan is often optimal, and two conjectures of Sun and Davis are true.
Clemm and Trebat-Leder (2014) proved that the number of quadratic number fields with absolute discriminant bounded by $x$ over which there exist elliptic curves with good reduction everywhere and rational $j$-invariant is $\gg x\log^{-1/2}(x)$. In this paper, we assume the $abc$-conjecture to show the sharp asymptotic $\sim cx\log^{-1/2}(x)$ for this number, obtaining formulae for $c$ in both the real and imaginary cases. Our method has three ingredients: (1) We make progress towards a conjecture of Granville: Given a fixed elliptic curve $E/\mathbb{Q}$ with short Weierstrass equation $y^2 = f…
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）
Let $f$ be a primitive form with respect to $SL_2(Z)$. Then we propose a conjecture on the congruence between the Klingen-Eisenstein lift of the Duke-Imamoglu-Ikeda lift of $f$ and a certain lift of a vector valued Hecke eigenform with respect to $Sp_2(Z)$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.
An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a conjecture of Erdos and Graham, that, for each fixed number K>1, the complement in the integers of any union of residue classes r(n) mod n, for distinct n in (N,KN], has density at least d_K for N sufficiently large. He…
It is known that Szpiro's conjecture, or equivalently the ABC-conjecture, implies Lang's conjecture giving a uniform lower bound for the canonical height of nontorsion points on elliptic curves. In this note we show that a significantly weaker version of Szpiro's conjecture, which we call "prime-depleted," suffices to prove Lang's conjecture.
A natural variant of Lehmer's conjecture that the Ramanujan $τ$-function never vanishes asks whether, for any given integer $α$, there exist any $n \in \mathbb{Z}^+$ such that $τ(n) = α$. A series of recent papers excludes many integers as possible values of the $τ$-function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for $τ(n)$. We synthesize these results and methods to prove that if $0 < |α| < 100$ and $α\notin T := \{2^k, -24,-48, -70,-90, 92, -96\}$, then $τ(n) \neq α$ for all $n > 1$. Moreover, if $α\in T$ and $τ(n) =…
Every polynomial $P(X)\in \mathbb Z[X]$ satisfies the congruences $P(n+m)\equiv P(n) \mod m$ for all integers $n, m\ge 0$. An integer valued sequence $(a_n)_{n\ge 0}$ is called a pseudo-polynomial when it satisfies these congruences. Hall characterized pseudo-polynomials and proved that they are not necessarily polynomials. A long standing conjecture of Ruzsa says that a pseudo-polynomial $a_n$ is a polynomial as soon as $\limsup_n \vert a_n\vert^{1/n}<e$. Under this growth assumption, Perelli and Zannier proved that the generating series $\sum_{n=0}^\infty a_n z^n$ is a $G$-function. A primar…
