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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: U0(P, Q) = 0, U1(P, Q) = 1, Un+1(P, Q) = P · Un(P, Q) − Q · Un−1(P, V0(P, Q) = 2, V1(P, Q) = P, Vn+1(P, Q) = P · Vn(P, Q) − Q · Vn−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 U0(1, −1) = 0 = 02, U1(1, −1) = 1 = 12, U12(1, −1) = 144 = 122. 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 · z2 + 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 z2 ± 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: Vn(P, Q)2 − D · Un(P, Q)2 = 4Qn, (1) where D = P2 − 4Q is discriminant. For |Q| = 1, it means that the pairs (Vn(P, Q), Un(P, Q)) form solutions to the Pellian equation (that does not involve n): x2 − Dy2 = ±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 = P2 − 4Q > 0. If positive integers u and v are such that v 2 − Du2 = ±4, then u = Un(P, Q) and v = Vn(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 y2 − 5x2 = ±4 for some integer y. This allows us to reduce finding Fibonacci numbers of the form x = az2 + b to solving the Diophantine equation: y2 = 5(az2 + b)2 ± 4 = 5a2z4 + 10abz2 + (5b2 ± 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 y2 = 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: y2 = ax4 + bx2 + c with irreducible right hand side firstly to a Diophantine equation x2 = 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

• n 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: y2 = ax4 + bx2 + c with integer coefficients a, b, c, ac = 0, b2 − 4ac = 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 4c and represent it as: (b2 − 4ac)(x2)2 + 4cy2 − (bx2 + 2c)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 (x0, y0, z0) with z0 = 0 be a particular non-trivial integer solution of the Diophantine equation Ax2 + By 2 + Cz2 = 0. The general integer solution to the above equation is given by (x, y, z) = p q (Px(m, n), Py(m, n), Pz(m, n)) (3) where m, n as well as p, q are coprime integers with q > 0 dividing 2 lcm(A, B)Cz2

0, and

Px(m, n) = x0Am2 + 2y0Bmn − x0Bn2, Py(m, n) = −y0Am2 + 2x0Amn + y0Bn2, Pz(m, n) = z0Am2 + z0Bn2.

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 x2 = f (m,n)

g(m,n)

Denoting X = x2, Y = y, Z = bx2 + 2c, A = b2 − 4ac, B = 4c, 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 · (Px(m, n), Py(m, n), Pz(m, n)) , where m, n are coprime integers, and r is a rational number. In our case, this solution should additionally satisfy the relation: 2c = Z − bX = r · (Pz(m, n) − b · Px(m, n)) , implying that r =

2c Pz(m,n)−b·Px(m,n). So we get a constraining Diophantine

equation: x2 = 2c · Px(m, n) Pz(m, n) − b · Px(m, n). (5)

Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Properties of the equation z2 = P1(x,y)

P2(x,y)

So our goal is to solve the equation: z2 = P1(x, y) P2(x, y) in integers (x, y, z) = (m, n, k) with gcd(m, n) = 1, where P1 and P2 are homogeneous quadratic polynomials with Disc(P1) = 0, Disc(P2) = 0, and Res(P1, P2) = 0. We start with the following theorem:

Theorem

Let P1(x, y) and P2(x, y) be homogeneous quadratic polynomials with integer coefficients and R = Res(P1, P2) = 0. Then there exists an integer G such that for any coprime integers m, n, gcd(P1(m, n), P2(m, n)) divides G.

In fact, the smallest such G equals the largest element in the Smith normal form of the resultant matrix of P1 and P2.

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 z2 = P1(x,y)

P2(x,y). Then by previous theorem,

gcd(P1(m, n), P2(m, n)) = P2(m, n) divides a certain integer G. Then for some (positive or negative) divisor g of G, we have P2(m, n) = g and P1(m, n) = gk2. So (x, y, z) = (m, n, k) represents a solution to the following system of equations:

• P1(x, y) = g · z2,

P2(x, y) = g. (6) Therefore, to find all solutions to z2 = P1(x,y)

P2(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

Solving the first equation

We start with solving the first equation of the system: P1(x, y) = g · z2 with the Trick and the Hammer.

Theorem

Any homogeneous quadratic polynomial with integer coefficients and non-zero discriminant can be represented as a linear combination with non-zero rational coefficients of squares of two homogeneous linear polynomials. Moreover, these polynomials are linearly independent. Proof is almost trivial: ax2 + bxy + cy 2 =          ax2 + cy 2, b = 0

1 4a ·

• (2ax + by)2 + (4ac − b2)y 2

, a = 0

1 4c ·

• (4ac − b2)x2 + (bx + 2cy)2

, c = 0

b 4 ·

• (x + y)2 − (x − y)2

, a = c = 0.

Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Reduction to a quartic equation

Thanks to the previous theorem, we can rewrite the first equation P1(x, y) = g · z2 as a · Q1(x, y)2 + b · Q2(x, y)2 − g · z2 = 0. (7) We solve this equation with our Hammer Theorem to obtain Q1(x, y) = p

q · R1(m, n) and Q2(x, y) = p q · R2(m, n), where q > 0 ranges

• ver the positive divisors of a certain integer and integer p is coprime to q.

We solve these linear equations with respect to x, y to obtain (x, y) = p

q · (Sx(m, n), Sy(m, n)) , where Sx(m, n) and Sy(m, n) are linear

homogeneous polynomials with rational coefficients. Plugging this into the second equation P2(x, y) = g, we get a finite number of quartic equations w.r.t. m, n: P2(Sx(m, n), Sy(m, n)) = gq2 p2 . (8)

Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Solving quartic equation

It remains to solve the equation: f (m, n) = d w.r.t. coprime integers m, n, where f is homogeneous quartic polynomial with Disc(f ) = 0. If f is irreducible, it is a Thue equation and we can employ PARI/GP to solve it. Otherwise f = g · h where g and h are polynomials of degree at most 3 and we solve the systems:

• g(m, n) = k,

h(m, n) = d

k ,

where k runs over the (positive and negative) divisors of d. Each system is solved as follows: if one of the factors, say, g is linear, then we solve g(m, n) = k to express m in terms of n. if both g and h are quadratic and irreducible, we form the homogeneous equation

d k · g(m, n) − k · h(m, n) = 0 and solve it w.r.t. m n .

Then we plug the expression of m in terms of n into the equation h(m, n) = d

k to obtain

easily solvable univariate equation.

Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Summary: two theorems

Theorem

Finding integral points on a biquadratic curve y 2 = a · x4 + b · x2 + c (9) with integer coefficients a, b, c, ac(b2 − 4ac) = 0, reduces to solving a finite number of quartic Thue equations.

Theorem

Let P1(x, y) = a1x2 + b1xy + c1y 2 and P2(x, y) = a2x2 + b2xy + c2y 2 be homogeneous quadratic polynomials with integer coefficients such that Disc(P1) = 0, Disc(P2) = 0, and Res(P1, P2) = 0. Then the equation z2 = P1(x, y) P2(x, y) (10) has a finite number of integer solutions (x, y, z) = (m, n, k) with gcd(m, n) = 1.

Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

One more result

Our method also allows one to find solutions to a system of Diophantine equations:

• a1x2 + c1z = d1,

b2y2 + c2z2 = d2. From this perspective, it continues our earlier work (MA, INTEGERS 2011), where we described how to compute solutions to a system of Diophantine equations:

• a1x2 + b1y2 + c1z2 = d1,

a2x2 + b2y2 + c2z2 = d2, and demonstrated applications for finding common terms of distinct Lucas sequences of the form U(P, ±1) or V (P, ±1), which include Fibonacci, Lucas, Pell, and Lucas-Pell numbers.

Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Near-squares in Lucas sequences

Theorem

For fixed integers a = 0 and b, finding terms of the form am2 + b in nondegenerate Lucas sequences U(P, Q) or V (P, Q) with Q = ±1 reduces to a finite number of Thue equations, unless it is V (P, Q) and b = ±2. Thus there are only a finite number of such terms in these Lucas sequences. Form Fibonacci numbers Lucas numbers Pell numbers Pell-Lucas numbers m2 + 1 1, 2, 5 1 1, 2, 5 2, 82 m2 − 1 0, 3, 8 3 none m2 + 2 2, 3 2, 11, and V4n+2 2 2 and V4n+2 m2 − 2 2, 34 V4n 2 14 and V4n m2 + 3 3 4, 199 12 none m2 − 3 1, 13, 1597 1 1 none 2m2 + 1 1, 3 1, 3 1 none 2m2 − 1 1 1, 7, 199 1 none 2m2 + 2 2, 34 2, 4 2 2 and V4n 2m2 − 2 none 0, 70 V4n+2 2m2 + 3 3, 5, 21 3, 11 5 none 2m2 − 3 5 29, 47, 64079 5, 29 none

Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences

Acknowledgements

• Dr. John Cremona, University of Warwick

The PARI/GP development group National Science Foundation grant no. IIS-1253614 In details the presented results are described in the papers: “On integral points on biquadratic curves and near multiples of squares in Lucas sequences”. http://arxiv.org/abs/1306.0883 “On the intersections of Fibonacci, Pell, and Lucas numbers”. INTEGERS 11(3) (2011), pp. 239-259. http://arxiv.org/abs/1002.1679

Max Alekseyev Integral points on biquadratic curves and near-multiples of squares in Lucas sequences