Integral points on biquadratic curves and near-multiples of squares - PowerPoint PPT Presentation

Integral points on biquadratic curves and near-multiples of squares in Lucas sequences Max Alekseyev Dept. Computer Science and Engineering 2013 Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Lucas sequences Lucas sequences U ( P , Q ) and V ( P , Q ) are defined by the same linear recurrent relation with the coefficient P , Q ∈ Z but different initial terms: U 0 ( P , Q ) = 0 , U 1 ( P , Q ) = 1 , U n +1 ( P , Q ) = P · U n ( P , Q ) − Q · U n − 1 ( P , V 0 ( P , Q ) = 2 , V 1 ( P , Q ) = P , V n +1 ( P , Q ) = P · V n ( P , Q ) − Q · V n − 1 ( P , Some Lucas sequences have their own names: Sequence Name Initial terms U (1 , − 1) Fibonacci numbers 0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , . . . V (1 , − 1) Lucas numbers 2 , 1 , 3 , 4 , 7 , 11 , 18 , 29 , . . . U (2 , − 1) Pell numbers 0 , 1 , 2 , 5 , 12 , 29 , 70 , 169 , . . . V (2 , − 1) Pell-Lucas numbers 2 , 2 , 6 , 14 , 34 , 82 , 198 , . . . Other examples include Jacobsthal numbers U (1 , − 2), Mersenne numbers U (3 , 2) etc. We focus on the case of Q = 1 or Q = − 1. Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Squares among Fibonacci numbers In 1964, Cohn and Wyler independently proved that the only squares among Fibonacci numbers are U 0 (1 , − 1) = 0 = 0 2 , U 1 (1 , − 1) = 1 = 1 2 , U 12 (1 , − 1) = 144 = 12 2 . Since then the question of finding (multiples of) squares and higher powers in Lucas sequences has been extensively studied. At the same time, finding near-squares, i.e., numbers of the form a · z 2 + b for fixed non-zero a , b , has got only limited attention. To the best of our knowledge, only Fibonacci and Lucas numbers of the form z 2 ± 1 are known due to Finkelstein (1973, 1975) and Robbins (1981). Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Characterization of Lucas sequences Terms of Lucas sequences U ( P , Q ) and V ( P , Q ) satisfy the identity: V n ( P , Q ) 2 − D · U n ( P , Q ) 2 = 4 Q n , (1) where D = P 2 − 4 Q is discriminant. For | Q | = 1, it means that the pairs ( V n ( P , Q ) , U n ( P , Q )) form solutions to the Pellian equation (that does not involve n ): x 2 − Dy 2 = ± 4 . (2) The converse statement can be used to prove that given positive integers belong to V ( P , Q ) or U ( P , Q ) respectively: Theorem Let P, Q be integers such that P > 0 , | Q | = 1 , ( P , Q ) � = (3 , 1) , and D = P 2 − 4 Q > 0 . If positive integers u and v are such that v 2 − Du 2 = ± 4 , then u = U n ( P , Q ) and v = V n ( P , Q ) for some integer n ≥ 0 . Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Near-squares in Fibonacci numbers For Fibonacci numbers U (1 , − 1), we have D = 5 and thus x is a Fibonacci number if and only if y 2 − 5 x 2 = ± 4 for some integer y . This allows us to reduce finding Fibonacci numbers of the form x = az 2 + b to solving the Diophantine equation: y 2 = 5( az 2 + b ) 2 ± 4 = 5 a 2 z 4 + 10 abz 2 + (5 b 2 ± 4) , which represents a biquadratic curve . Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Integral points on algebraic curves In 1929, Siegel proved that any equation y 2 = f ( x ) with irreducible polynomial f ∈ Z [ x ] of degree at least 3 has finitely many integer solutions. In 1966, Baker developed a method for bounding the solutions and performing an exhaustive search. For third-degree curves, Baker’s method was a subject to many practical improvements, and now there exist several software implementations for finding integral points on elliptic curves, including eclib , SAGE , MAGMA . However we are unaware of public software implementations of Baker’s method (and its practical feasibility) for curves of higher degree besides some special quartic curves addressed in MAGMA . Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Thue equations Thue equations of the form g ( x , y ) = d , where g is homogeneous irreducible polynomial of degree at least 3 were first studied by Thue in 1909, who proved that they have only a finite number of solutions. In computer era, Thue equations became a subject to developments of computational methods, resulting in at least two implementations: in computer algebra systems MAGMA and PARI/GP . For our practical computations, we chose freeware PARI/GP , whose Thue equations solver is based on Bilu and Hanrot’s improvement of Tzanakis and de Weger’s method. Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Our contribution We will show how to reduce a search for integral points on a biquadratic curve: y 2 = ax 4 + bx 2 + c with irreducible right hand side firstly to a Diophantine equation x 2 = f ( m , n ) g ( m , n ) in coprime integers m , n with homogeneous quadratic polynomials f and g , and then to a finite number of Thue equations. From practical perspective, this reduction enables finding integral points on biquadratic curves (and thus near multiples of squares in Lucas sequences), using readily available Thue equation solvers. From theoretical perspective, while possibility of reduction to Thue equations was described by Mordell in 1969 based on arguments from algebraic number theory, to the best of our knowledge, there is no simple algorithm applicable for the general case. In contrast to traditional treatment of this kind of problems, our reduction method is elementary. Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

The Trick Suppose we want to find integral points on a biquadratic curve: y 2 = ax 4 + bx 2 + c with integer coefficients a , b , c , ac � = 0, b 2 − 4 ac � = 0. The Trick : represent the equation as a linear combination of three squares with non-zero coefficients. In this case, we multiply the curve equation by 4 c and represent it as: ( b 2 − 4 ac )( x 2 ) 2 + 4 cy 2 − ( bx 2 + 2 c ) 2 = 0 . Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

The Hammer Theorem (MA, INTEGERS 2011) Let A , B , C be non-zero integers and let ( x 0 , y 0 , z 0 ) with z 0 � = 0 be a particular non-trivial integer solution of the Diophantine equation Ax 2 + By 2 + Cz 2 = 0 . The general integer solution to the above equation is given by ( x , y , z ) = p q ( P x ( m , n ) , P y ( m , n ) , P z ( m , n )) (3) where m , n as well as p , q are coprime integers with q > 0 dividing 2 lcm ( A , B ) Cz 2 0 , and x 0 Am 2 + 2 y 0 Bmn − x 0 Bn 2 , P x ( m , n ) = − y 0 Am 2 + 2 x 0 Amn + y 0 Bn 2 , P y ( m , n ) = z 0 Am 2 + z 0 Bn 2 . P z ( m , n ) = This theorem corrects an error in Corollary 6.3.8 of the classic Cohen 2007 book. Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Reduction to the equation x 2 = f ( m , n ) g ( m , n ) Denoting X = x 2 , Y = y , Z = bx 2 + 2 c , A = b 2 − 4 ac , B = 4 c , C = − 1, we get a Diophantine equation: A · X 2 + B · Y 2 + C · Z 2 = 0 . (4) Its general solution is given by: ( X , Y , Z ) = r · ( P x ( m , n ) , P y ( m , n ) , P z ( m , n )) , where m , n are coprime integers, and r is a rational number. In our case, this solution should additionally satisfy the relation: 2 c = Z − bX = r · ( P z ( m , n ) − b · P x ( m , n )) , 2 c implying that r = P z ( m , n ) − b · P x ( m , n ) . So we get a constraining Diophantine equation: 2 c · P x ( m , n ) x 2 = (5) P z ( m , n ) − b · P x ( m , n ) . Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Properties of the equation z 2 = P 1 ( x , y ) P 2 ( x , y ) So our goal is to solve the equation: z 2 = P 1 ( x , y ) P 2 ( x , y ) in integers ( x , y , z ) = ( m , n , k ) with gcd( m , n ) = 1, where P 1 and P 2 are homogeneous quadratic polynomials with Disc ( P 1 ) � = 0, Disc ( P 2 ) � = 0, and Res ( P 1 , P 2 ) � = 0. We start with the following theorem: Theorem Let P 1 ( x , y ) and P 2 ( x , y ) be homogeneous quadratic polynomials with integer coefficients and R = Res ( P 1 , P 2 ) � = 0 . Then there exists an integer G such that for any coprime integers m , n, gcd( P 1 ( m , n ) , P 2 ( m , n )) divides G. In fact, the smallest such G equals the largest element in the Smith normal form of the resultant matrix of P 1 and P 2 . Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

To a system quadratic equations Suppose that ( x , y , z ) = ( m , n , k ) with gcd( m , n ) = 1 satisfies the equation z 2 = P 1 ( x , y ) P 2 ( x , y ) . Then by previous theorem, gcd( P 1 ( m , n ) , P 2 ( m , n )) = P 2 ( m , n ) divides a certain integer G . Then for some (positive or negative) divisor g of G , we have P 2 ( m , n ) = g and P 1 ( m , n ) = gk 2 . So ( x , y , z ) = ( m , n , k ) represents a solution to the following system of equations: � P 1 ( x , y ) = g · z 2 , (6) P 2 ( x , y ) = g . Therefore, to find all solutions to z 2 = P 1 ( x , y ) P 2 ( x , y ) , we need to solve the above systems for g ranging over the divisors of G . Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences