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A GENERALIZATION OF SIEGEL’S THEOREM AND HALL’S CONJECTURE
"... Abstract. Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is placed upon the number of prime factors dividing a ..."
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Abstract. Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is placed upon the number of prime factors dividing a fixed coordinate? If the bound is zero, then Siegel’s Theorem guarantees that there are only finitely many such points. We consider, theoretically and computationally, two conjectures: one is a generalization of Siegel’s Theorem and the other is a refinement which resonates with Hall’s conjecture. 1.
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 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 ∞.