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**1 - 3**of**3**### On Hall’s conjecture

"... We show that for any even positive integer δ there exist polynomials x and y with integer coefficients such that deg(x) = 2δ, deg(y) = 3δ and deg(x 3 − y 2) = δ + 5. Hall’s conjecture asserts that for any ε> 0, there exists a constant c(ε)> 0 such that if x and y are positive integers satisfying ..."

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We show that for any even positive integer δ there exist polynomials x and y with integer coefficients such that deg(x) = 2δ, deg(y) = 3δ and deg(x 3 − y 2) = δ + 5. Hall’s conjecture asserts that for any ε> 0, there exists a constant c(ε)> 0 such that if x and y are positive integers satisfying x3 − y2 ̸ = 0, then |x3 − y2 |> c(ε)x1/2−ε. It is known that Hall’s conjecture follows from the abc-conjecture. For a stronger version of Hall’s conjecture which is equivalent to the abc-conjecture see [3, Ch. 12.5]. Originally, Hall [8] conjectured that there is C> 0 such that |x3 − y2 | ≥ C √ x for positive integers x, y with x3 − y2 ̸ = 0, but this formulation is unlikely to be true. Danilov [4] proved that 0 < |x3 −y 2 | < 0.97 √ x has infinitely many solutions in positive integers x, y; here 0.97 comes from 54 √ 5/125. For examples with “very small ” quotients |x3 − y2 | / √ x, up to 0.021, see [7] and [9]. It is well known that for non-constant complex polynomials x and y, such that x3 ̸ = y2, we have deg(x3 − y2) / deg(x)> 1/2. More precisely, Davenport [6] proved that for such polynomials the inequality deg(x 3 − y 2) ≥ 1 2 deg(x) + 1 (1) holds. This statement also follows from Stothers-Mason’s abc theorem for polynomials (see, e.g., [10, Ch. 4.7]). Zannier [12] proved that for any positive integer δ there exist complex polynomials x and y such that deg(x) = 2δ, deg(y) = 3δ and x, y satisfy the equality in Davenport’s bound (1). In his previous paper [11], he related the existence of such examples with coverings of the Riemann sphere, unramified except above 0, 1 and ∞.

### Minimum Degree of the Difference of Two Polynomials over Q. Part II: Davenport–Zannier triples

, 2013

"... This article is sequel to our earlier article [8]. In [8], we proved the existence of certain polynomials with extremal properties defined over Q, but did not present the polynomials themselves. We do it here, in this paper. What is missing in this “zeroth draft ” is proofs. We, nevertheless, decide ..."

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This article is sequel to our earlier article [8]. In [8], we proved the existence of certain polynomials with extremal properties defined over Q, but did not present the polynomials themselves. We do it here, in this paper. What is missing in this “zeroth draft ” is proofs. We, nevertheless, decided to make this text available to our potential readers who might be curious about polynomials in question. 1