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On Hall's conjecture and related polynomial Pell's equation

Hall's conjecture asserts that for any epsilon >0, there exists a constant c(epsilon)>0 such that if x and y are positive integers satisfying x^3-y^2 <> 0, then |x^3-y^2| > c(epsilon) x^(1/2-epsilon). It is known that Hall's conjecture follows from the abc-conjecture. Danilov proved that 0<|x^3-y^2| < 0.97 sqrt(x) has infinitely many solutions in positive integers x, y. Davenport proved that for non-constant complex polynomials x and y, such that x^3 <> y^2, the inequality deg(x^3-y^2) >= 1/2 deg(x) + 1 (*) holds. This statement also follows from Stothers-Mason's abc theorem for polynomials. Zannier proved that for any positive integer delta there exist complex polynomials x and y such that deg(x)=2 delta, deg(y)=3 delta and x, y satisfy the equality in Davenport's bound. It is natural to ask whether examples with the equality in (*) exist for polynomials with integer (rational) coefficients. Such examples are known only for delta=1, 2, 3, 4, 5. The examples for delta=5 were found by Birch, Chowla, Hall, Schinzel and Elkies. In these examples we have deg(x^3 - y^2) / deg(x) = 0.6, and it seems that no examples of polynomials with integer coefficients, satisfying x^3-y^2 <> 0 and deg(x^3 - y^2) / deg(x) < 0.6, were known. In this talk we will present our recent result which gives an explicit construction of integer polynomials x and y of arbitrarily large degrees with deg (x^3-y^2)- 1/2 deg (x) bounded from above by an absolute constant. For any epsilon>0 there exist polynomials x and y with integer coefficients such that x^3 <> y^2 and deg(x^3 - y^2) / deg(x) < 1/2 + epsilon. \emph{; ; ; More precisely, for any even positive integer delta there exist polynomials x and y with integer coefficients such that deg(x)=2 delta, deg(y)=3 delta and deg(x^3-y^2)=delta+5. The construction is based on the binary recursive sequence of polynomials given by a_1=0, a_2=t^2+1, a_m = 2ta_{; ; ; m-1}; ; ; +a_{; ; ; m-2}; ; ; . We give also an interpretation of our construction in terms of solutions of polynomial Pell's equation X^2 - (t^2+1)Y^2 = -1.

Hall's conjecture ; integer polynomials

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