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A NEW ALGORITHM TO SEARCH FOR SMALL NONZERO x 3 − y 2  VALUES
"... Abstract. In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero k = x 3 −y 2  values. Seventeen new values of k<x 1/2 are reported. 1. Hall’s conjecture Dealing with natural numbers, the difference (1.1) k = x 3 − y 2 is zero when x = t 2 and y = t 3 but, in ..."
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Abstract. In relation to Hall’s conjecture, a new algorithm is presented to search for small nonzero k = x 3 −y 2  values. Seventeen new values of k<x 1/2 are reported. 1. Hall’s conjecture Dealing with natural numbers, the difference (1.1) k = x 3 − y 2 is zero when x = t 2 and y = t 3 but, in other cases, it seems difficult to achieve small absolute values. For a given k ̸ = 0, (1.1), known as Mordell’s equation, is an elliptic curve and has only finitely many solutions in integers by Siegel’s theorem. Therefore, for any nonzero k value, there are only finitely many solutions in x (which is hence bounded). There is a proven lower bound, due to A. Baker [1] and improved by H. M. Stark [14], that places the size of k above the order of log c (x) for any c<1. A bound concerning the minimal growth rate of k  was found early by M. Hall [2, 7] by means of a parametric family of the form (1.2) f(t) = t 9 (t9 +6t 6 +15t 3 + 12), g(t) = t15 27 + t12 +4t9 +8t6 3 f 3 (t) − g2 (t) = − 3t6 +14t3+27
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 satis ..."
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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 abcconjecture. For a stronger version of Hall’s conjecture which is equivalent to the abcconjecture 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 nonconstant 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 StothersMason’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 ∞.