On Integral Points on Biquadratic Curves and Near-Multiples of Squares in Lucas Sequences

Max A. Alekseyev Department of Mathematics The George Washington University 2115 G St., NW Washington, DC 20052 USA

                     maxal@gwu.edu

Szabolcs Tengely Mathematical Institute University of Debrecen 4010 Debrecen, PO Box 12 Hungary

               tengely@science.unideb.hu

Abstract

We describe an algorithmic reduction of the search for integral points on a curve y^{2} = ax^{4} + bx^{2} + c with ac(b^{2} − 4ac) ≠ 0 to solving a finite number of Thue equations. While the existence of such a reduction is anticipated from arguments of algebraic number theory, our algorithm is elementary and is, to the best of our knowledge, the first published algorithm of this kind. In combination with other methods and powered by existing Thue equation solvers, it allows one to efficiently compute integral points on biquadratic curves.

We illustrate this approach with a particular application of finding near-multiples of squares in Lucas sequences. As an example, we establish that among Fibonacci numbers only 2 and 34 are of the form 2m^{2} + 2; only 1, 13, and 1597 are of the form m^{2} − 3; and so on.

As an auxiliary result, we also give an algorithm for solving a Diophantine equation k^{2} = \frac{f(m,n)}{g(m,n)} in integers m, n, k, where f and g are homogeneous quadratic polynomials.

1 Introduction

Siegel [26] proved that any equation y^{2} = f(x) with irreducible polynomial f ∈ ℤ[x] of degree at least 3 has finitely many integral points. With the method of Baker [2, 3, 4], it became possible to bound the solutions and perform an exhaustive search. For third-degree curves, Baker’s method was a subject to many practical improvements, and now there exists a number of software implementations for finding integral points on elliptic curves [6, 15, 27]. These procedures are based on a method developed by Stroeker and Tzanakis [29] and independently by Gebel, Pethő, and Zimmer [14].

Thue equations of the form g(x, y) = d, where g ∈ ℤ[x, y] is a homogeneous irreducible polynomial of degree at least 3 and d ∈ ℤ, were first studied by Thue [32], who proved that they have only a finite number of integer 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 [6] and PARI/GP [31]. For our practical computations, we chose SAGE [27], which adopts PARI/GP Thue equations solver based on Bilu and Hanrot’s improvement [5] of Tzanakis and de Weger’s method [33].

In the current work, we show how to reduce a search for integral points on a biquadratic curve

y^{2} = ax^{4} + bx^{2} + c

with integer (or, more generally, rational) coefficients a, b, c with ac(b^{2} − 4ac) ≠ 0 firstly to a Diophantine equation k^{2} = \frac{f(m,n)}{g(m,n)} in coprime integers m, n with homogeneous quadratic polynomials f and g (Theorem 7), and then to a finite number of quartic Thue equations (Theorem 5).^{1} While possibility of reduction to Thue equations was described by Mordell [18] based on arguments from algebraic number theory, to the best of our knowledge, there is no published algorithm applicable for the general case. Furthermore, in contrast to traditional treatment of this kind of problems with algebraic number theory [18, 28, 11, 21], our reduction method is elementary. It may be viewed as a generalization of the one of Steiner and Tzanakis [28] who reduced Ljunggren equation y^{2} = 2x^{4} − 1 to two Thue equations (in particular, we obtain the same Thue equations).

There are other methods which can be used in certain cases to determine all integral solutions of the equation y^{2} = ax^{4} + bx^{2} + c. Poulakis [22] provided an elementary algorithm to solve Diophantine equations of the form y^{2} = f(x), where f(x) is quartic monic polynomial with integer coefficients, based on Runge’s method [24, 25, 34]. Here it is crucial that the leading coefficient of f(x) is 1 (the idea also works if the leading coefficient is a perfect square). Using the theory of Pell equations, Kedlaya [16] described a method to solve the system of equations

1In the course of this reduction, we may also encounter Thue-like equations g(x, y) = d with g being a reducible homogeneous polynomial of degree 4. Such equations however are easily solvable [1, Theorem 3].

\begin{cases}x^{2} − a_{1}y^{2} = b_{1}, \\ P (x, y) = z^{2}, \end{cases}

where P is a given integer polynomial, and implemented his algorithm in Mathematica.^{2} If we set P (x, y) = c_{1}x + d_{1}, then we obtain a quartic equation of the form (a_{1}c_{1}y)^{2} = a_{1}z^{4} − 2a_{1}d_{1}z^{2} − a_{1}b_{1}c_{1}^{2}. There is also a simple reduction of the equation y^{2} = ax^{4} + bx^{2} + c to an elliptic equation: after multiplying the equation by a^{2}x^{2}, one obtains

(axy)^{2} = (ax^{2})^{3} + b(ax^{2})^{2} + ac(ax^{2}),

which can be further written as

Y^{2} = X^{3} + bX^{2} + acX.

As we noted earlier to determine all integral points on a given elliptic curve one can follow a method developed by Stroeker and Tzanakis [29] and independently by Gebel, Pethő and Zimmer [14]. The disadvantage of this approach is that there is no known algorithm to determine the rank of the so-called Mordell-Weil group of an elliptic curve, which is necessary to determine all integral points on the curve.

For efficient computation of integral points on biquadratic curves, we implemented in SAGE a combination of the elliptic curve and reduction to Thue equations methods.^{3} By default we employ the elliptic curve method and if it fails, we fall back to our reduction to Thue equations.

Our approach also allows one to efficiently compute solutions to a system of Diophantine equations (Theorem 8):

\begin{cases}a_{1}x^{2} + c_{1}z = d_{1}, \\ b_{2}y^{2} + c_{2}z^{2} = d_{2}. \end{cases}

From this perspective, it continues earlier work [1], where the first author described an algorithm for computing solutions to a system of Diophantine equations:

\begin{cases}a_{1}x^{2} + b_{1}y^{2} + c_{1}z^{2} = d_{1}, \\ a_{2}x^{2} + b_{2}y^{2} + c_{2}z^{2} = d_{2}, \end{cases}

and demonstrated applications for finding common terms of distinct Lucas sequences of the form U(P, ±1) or V (P, ±1), which include the Fibonacci, Lucas, Pell, and Lucas-Pell numbers. The current method also has applications for such Lucas sequences, allowing one to find all terms of the form a · m^{2} + b for any fixed integers a, b. While the question of finding multiples of squares (i.e., with b = 0) in Lucas sequences has been widely studied, starting with the works of Cohn [9] and Wyler [36] (we refer to Bremner and Tzanakis [7] for an extensive review of the literature), finding near multiples of squares (i.e., with b ≠ 0) has got so far only a limited attention [12, 13, 23, 35]. In the current work, we present an unified computational approach for solving this problem. As an example, we establish that among Fibonacci numbers only 2 and 34 are of the form 2m^{2} + 2; only 1, 13, and 1597 are of the form m^{2} − 3; and so on.

2Kedlaya’s implementation is available from http://math.ucsd.edu/~kedlaya/papers/pell.tar.

3Our implementation is available from http://www.math.unideb.hu/~tengely/biquadratic.sage.

The paper is organized as follows. In Section 2, we develop our machinery for homogeneous quadratic polynomials with integer coefficients. In Section 3, we prove our method for finding integral points on biquadratic curves and illustrate its workflow on Ljunggren equation. In Section 4, we further demonstrate how our method can be used for finding near-multiples of squares in Lucas sequences and list some results of this kind.

2 Homogeneous quadratic polynomials

We start with studying properties of quadratic homogeneous polynomials with integer coefficients in two and three variables. We do not distinguish between homogeneous polynomials in two variables from their univariate counterparts (i.e., f(x, y) = ax^{2} + bxy + cy^{2} and f̃(z) = az^{2} + bz + c) that allows us to define resultant (Res) and discriminant (Disc) on them.

Theorem 1 (Theorem 5 in [1]^{4}). 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 to the Diophantine equation Ax^{2} + By^{2} + Cz^{2} = 0. Then its general integer solution is given by

(x, y, z) = \frac{p}{q} (P_{x}(m, n), P_{y}(m, n), P_{z}(m, n))  (1)

where m, n as well as p, q are coprime integers with q > 0 dividing 2 lcm(A, B)Cz_{0}^{2}, and

\begin{aligned}P_{x}(m, n) & = & x_{0}Am^{2} + 2y_{0}Bmn − x_{0}Bn^{2}, \\ P_{y}(m, n) & = & −y_{0}Am^{2} + 2x_{0}Amn + y_{0}Bn^{2}, \\ P_{z}(m, n) & = & z_{0}Am^{2} + z_{0}Bn^{2}. \end{aligned}  (2)

We refer to [8, 10] for general methods of finding a particular solution to a quadratic homogeneous equation in three variables.^{5}

Theorem 2. 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. Let G be the largest element in the Smith normal form of the resultant matrix of P_{1} and P_{2}.^{6} Then for any coprime integers m, n, gcd(P_{1}(m, n), P_{2}(m, n)) divides G.

4This theorem corrects an error in [8, Corollary 6.3.8].

5In PARI/GP, a particular solution can be computed with the function bnfisnorm.

Proof. Let P_{1}(x, y) = a_{1}x^{2}+b_{1}xy+c_{1}y^{2} and P_{2}(x, y) = a_{2}x^{2}+b_{2}xy+c_{2}y^{2} where a_{1}, b_{1}, c_{1}, a_{2}, b_{2}, c_{2} are integer. Consider polynomials x · P_{1}(x, y), y · P_{1}(x, y), x · P_{2}(x, y), y · P_{2}(x, y) as linear combinations with integer coefficients of the basis terms x^{3}, x^{2}y, xy^{2}, y^{3}. Our goal is to find two linear combinations of these polynomials: one equal an integer multiple of x^{3} and the other equal an integer multiple of y^{3}. This corresponds to the following two systems of linear equations with the same resultant matrix:

\begin{matrix}x^{3} : \\ x^{2}y : \\ xy^{2} : \\ y^{3} : \end{matrix}  [\begin{matrix}a_{1} & 0 & a_{2} & 0 \\ b_{1} & a_{1} & b_{2} & a_{2} \\ c_{1} & b_{1} & c_{2} & b_{2} \\ 0 & c_{1} & 0 & c_{2} \end{matrix}] · [\begin{matrix}t_{1} \\ t_{2} \\ t_{3} \\ t_{4} \end{matrix}] = [\begin{matrix}1 \\ 0 \\ 0 \\ 0 \end{matrix}] or [\begin{matrix}0 \\ 0 \\ 0 \\ 1 \end{matrix}]  (3)

with respect to rational numbers t_{1}, t_{2}, t_{3}, t_{4}. Since the matrix determinant equals the resultant R ≠ 0, the systems have unique solutions of the form:

(t_{1}, t_{2}, t_{3}, t_{4}) = \frac{1}{R} (Δ_{1}, Δ_{2}, Δ_{3}, Δ_{4}) = \frac{1}{G} (\frac{Δ_{1}}{d_{3}} , \frac{Δ_{2}}{d_{3}} , \frac{Δ_{3}}{d_{3}} , \frac{Δ_{4}}{d_{3}} )

where each Δ_{i} is the determinant of a certain 3 × 3 minor of the resultant matrix and d_{3} is its third determinant divisor. Here we used the fact that \frac{R}{d_{3}} = G is the fourth elementary divisor of the resultant matrix (and the largest element of its Smith normal form). It is important to notice that all vector components \frac{Δ_{i}}{d_{3}} are integer.

So we have two linear combinations of x·P_{1}(x, y), y·P_{1}(x, y), x·P_{2}(x, y), y·P_{2}(x, y) with integer coefficients \frac{Δ_{1}}{d_{3}} , \frac{Δ_{2}}{d_{3}} , \frac{Δ_{3}}{d_{3}} , \frac{Δ_{4}}{d_{3}} equal G·x^{3} and G·y^{3}, respectively. For (x, y) = (m, n) where m, n are coprime integers, these linear combinations imply that gcd(P_{1}(m, n), P_{2}(m, n)) divides both G·m^{3} and G·n^{3}. Therefore, gcd(P_{1}(m, n), P_{2}(m, n)) divides gcd(G·m^{3}, G·n^{3}) = G · gcd(m^{3}, n^{3}) = G.

Remark 3. To compute G in practice, we do not need to compute Smith normal form of the resultant matrix. Instead, we simply solve the two linear systems (3) and define G as the least common multiple of all denominators in both solutions (t_{1}, t_{2}, t_{3}, t_{4}).

Theorem 4. 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. Let ax^{2} + bxy + cy^{2} be a homogeneous quadratic polynomial with integer coefficients and non-zero discriminant, i.e., b^{2} − 4ac ≠ 0. If b = 0 then ac ≠ 0 and the statement is trivial.

6While G = R would also satisfy the theorem statement, we want G as small as possible. The resultant R is often much larger than G defined in the theorem.

Suppose that b ≠ 0. If a ≠ 0, then we have

ax^{2} + bxy + cy^{2} = \frac{1}{4a} · (2ax + by)^{2} + \frac{4ac − b^{2}}{4a} · y^{2}.

Similarly, if c ≠ 0, then we have

ax^{2} + bxy + cy^{2} = \frac{4ac − b^{2}}{4c} · x^{2} + \frac{1}{4c} · (bx + 2cy)^{2}.

Finally, if a = c = 0, then

bxy = \frac{b}{4} · (x + y)^{2} − \frac{b}{4} · (x − y)^{2}.

It is easy to see that in all cases, the linear polynomials are linearly independent.

Theorem 5. Let P_{1}(x, y) and P_{2}(x, y) be homogeneous quadratic polynomials with integer coefficients such that Disc(P_{1}) ≠ 0, Disc(P_{2}) ≠ 0, and Res(P_{1}, P_{2}) ≠ 0. Then the equation

z^{2} = \frac{P_{1}(x, y)}{P_{2}(x, y)}  (4)

has a finite number of integer solutions (x, y, z) = (m, n, k) with gcd(m, n) = 1.

Proof. Suppose that (x, y, z) = (m, n, k) with gcd(m, n) = 1 satisfies the equation (4). Since P_{2}(m, n) divides P_{1}(m, n), we have gcd(P_{1}(m, n), P_{2}(m, n)) = P_{2}(m, n), which by Theorem 2 must divide a certain integer G. Then for some divisor^{7} 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:

\begin{cases}P_{1}(x, y) = g · z^{2}, \\ P_{2}(x, y) = g. \end{cases}  (5)

Therefore, to find all solutions to (4), we need to solve the systems (5) for g ranging over the divisors of G. Each such system is solved as follows.

We start with using Theorem 4 to represent P_{1}(x, y) as a linear combination with rational coefficients of squares of two linearly independent polynomials, say, P_{1}(x, y) = a·Q_{1}(x, y)^{2}+b· Q_{2}(x, y)^{2}. Substituting this representation into the first equation of (5), we get the following equation:

a · Q_{1}(x, y)^{2} + b · Q_{2}(x, y)^{2} − g · z^{2} = 0.  (6)

We solve this equation using Theorem 1 to obtain Q_{1}(x, y) = \frac{p}{q} · R_{1}(m, n) and Q_{2}(x, y) = \frac{p}{q} · R_{2}(m, n), where q ranges over the positive divisors of a certain integer and integer p is coprime to q. We solve this system of linear equations with respect to x, y to obtain

(x, y) = \frac{p}{q} · (S_{x}(m, n), S_{y}(m, n)) ,

7Unless specified otherwise, the divisors of an integer include both positive and negative divisors.

where S_{x}(m, n) and S_{y}(m, n) are linear homogeneous polynomials with rational coefficients (which depend only on g but not p, q). Plugging this into the second equation of (5), we get the following quartic equations with integer coefficients w.r.t. m, n:

ℓ_{g} · P_{2}(S_{x}(m, n), S_{y}(m, n)) = q^{2} · \frac{g · ℓ_{g}}{p^{2}} .  (7)

where ℓ_{g} is the least common multiple of the coefficients denominators of P_{2}(S_{x}(m, n), S_{y}(m, n)) (notice that ℓ_{g} depends only on g but not p, q). Here p^{2} must divide gℓ_{g} and thus there is a finite number of such equations. By [1, Theorem 3], each such equation has a finite number of solutions, unless P_{2}(S_{x}(m, n), S_{y}(m, n)) = c · T (m, n)^{2} for some polynomial T (x, y), which is not the case since Disc(P_{2}) ≠ 0.

Remark 6. Different choices of values for g, p, q may result in the same equation (7). In particular, if g′ = g · d^{2} for some integer d, then we can represent equation (6) for g′ in the form a · Q_{1}(x, y)^{2} + b · Q_{2}(x, y)^{2} − g · (d · z)^{2} = 0 so that it has the same solutions w.r.t. Q_{1}(x, y) and Q_{2}(x, y). Then the equation (7) for g′ has the same left hand side as the one for g (with ℓ_{g′} = ℓ_{g}), while the former has an extra factor d^{2} in the right hand side. Therefore, to reduce the number of equations in practice, we can restrict g to the square-free divisors of G: for each such g, we compute the left hand side of (7) and iterate over all distinct integer right hand sides of the form q^{2} · \frac{g·ℓ_{g}·d^{2}}{p^{2}} , where d^{2} divides \frac{G}{g}.

3 Finding integral points on biquadratic curves

Now we are ready to prove our main result:

Theorem 7. Finding integral points on a biquadratic curve

y^{2} = a · x^{4} + b · x^{2} + c  (8)

with integer coefficients a, b, c and ac(b^{2} − 4ac) ≠ 0, reduces to solving a finite number of quartic Thue equations.

Proof. Multiplying the equation (8) by 4c, we can rewrite it as a linear combination of three squares with non-zero integer coefficients:^{8}

(b^{2} − 4ac)(x^{2})^{2} + 4cy^{2} − (bx^{2} + 2c)^{2} = 0.

Denoting X = x^{2}, Y = y, Z = bx^{2} +2c, A = b^{2} −4ac, B = 4c, C = −1, we get a Diophantine equation:

A · X^{2} + B · Y^{2} + C · Z^{2} = 0.  (9)

8Alternatively, we can multiply (8) by 4a and obtain another linear combination of three squares: (2ax^{2} + b)^{2} + (4ac − b^{2}) · 1^{2} − 4ay^{2} = 0, which has smaller coefficients and thus may be preferable when c ≫ a.

If this equation is solvable with a particular solution (X, Y, Z) = (X_{0}, Y_{0}, Z_{0}), Z_{0} ≠ 0, then by Theorem 1 its general solution is given by

(X, Y, Z) = r · (P_{x}(m, n), P_{y}(m, n), P_{z}(m, n)) ,

where P_{i}(m, n) are polynomials defined by (2), 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 · (P_{z}(m, n) − b · P_{x}(m, n)) ,

implying that

r = \frac{2c}{P_{z}(m, n) − b · P_{x}(m, n)}.

So we get a constraining Diophantine equation:

x^{2} = \frac{2c · P_{x}(m, n)}{P_{z}(m, n) − b · P_{x}(m, n)},  (10)

which reduces to a finite number of Thue equations by Theorem 5, unless Res(P_{x}, P_{z}−b·P_{x}) = 0 or Disc(P_{z} − b · P_{x}) = 0.

The case of Res(P_{x}, P_{z}−b·P_{x}) = Res(P_{x}, P_{z}) = 0 is impossible since by direct computation we have Res(P_{x}, P_{z}) = −4AB^{2}CZ_{0}^{4} ≠ 0.

In the case of Disc(P_{z} − b · P_{x}) = 0, we have Disc(P_{z} − b · P_{x}) = 4b^{2}B^{2}Y_{0}^{2} − 4AB(Z_{0}^{2} − b^{2}X_{0}^{2}) = 0 and thus b^{2}BY_{0}^{2} = A(Z_{0}^{2} − b^{2}X_{0}^{2}). Since BY_{0}^{2} = −AX_{0}^{2} − CZ_{0}^{2}, we further get b^{2}(−AX_{0}^{2} −CZ_{0}^{2}) = A(Z_{0}^{2} −b^{2}X_{0}^{2}), which reduces to A+b^{2}C = 0. However this is impossible since A + b^{2}C = −4ac ≠ 0.

Therefore, Theorem 5 is applicable.

As a corollary we get

Theorem 8. The system of Diophantine equations:

\begin{cases}z = ax^{2} + d_{1}, \\ z^{2} = by^{2} + d_{2}, \end{cases}

where a, b, d_{1}, d_{2} are rational numbers with abd_{2}(d_{1}^{2} − d_{2}) ≠ 0, reduces to a finite number of quartic Thue equations.

Proof. The system implies that (ax^{2} +d_{1})^{2} = by^{2} +d_{2} or (by)^{2} = a^{2}bx^{4} +2abd_{1}x^{2} +b(d_{1}^{2} −d_{2}). Since (2d_{1}ab)^{2} − 4a^{2}b^{2}(d_{1}^{2} − d_{2}) = 4a^{2}b^{2}d_{2} ≠ 0, Theorem 7 applies.

3.1 Example: Ljunggren equation

We illustrate our method on Ljunggren equation y^{2} = 2x^{4} − 1 whose reduction to Thue equations was first obtained in [28]. Here we have (a, b, c) = (2, 0, −1). First we compute A = b^{2} − 4ac = 8, B = 4c = −4, and C = −1 and consider equation (9). Its particular solution is (1, 1, 2), which yields by Theorem 1 a general solution:

(X, Y, Z) = \frac{p}{q} (P_{x}(m, n), P_{y}(m, n), P_{z}(m, n))

with

P_{x}(m, n) = 8m^{2} − 8mn + 4n^{2},
P_{y}(m, n) = −8m^{2} + 16mn − 4n^{2},
P_{z}(m, n) = 16m^{2} − 8n^{2}.

Now we consider equation (10):

x^{2} = \frac{2cP_{x}(m, n)}{P_{z}(m, n) − b · P_{x}(m, n)} = \frac{−2(8m^{2} − 8mn + 4n^{2})}{16m^{2} − 8n^{2}} = \frac{−2m^{2} + 2mn − n^{2}}{2m^{2} − n^{2}}

and use Theorem 5 to solve it. We take the resultant matrix of P_{1}(x, y) = −2x^{2} + 2xy − y^{2} and P_{2}(x, y) = 2x^{2} − y^{2} and solve the two linear systems (3) to obtain (t_{1}, t_{2}, t_{3}, t_{4}) = \frac{1}{4}(−2, −1, 0, 1) and (t_{1}, t_{2}, t_{3}, t_{4}) = (−1, −1, −1, 0). So we get G = 4. Let g range over the divisors of G, i.e., g ∈ {−4, −2, −1, 1, 2, 4}.

We use Theorem 4 to represent P_{1}(x, y) as a linear combination of squares:

P_{1}(x, y) = −2x^{2} + 2xy − y^{2} = −\frac{1}{8} · (−4x + 2y)^{2} − \frac{1}{2} · y^{2} = −\frac{1}{2} · (−2x + y)^{2} − \frac{1}{2} · y^{2}

and obtain the equation (6) (multiplied by −2):

(−2x + y)^{2} + y^{2} + 2gz^{2} = 0.

Clearly, it may have non-trivial solutions only for g < 0 and only when −1 is a quadratic residue modulo 2g, which leaves us the only suitable value g = −1. The corresponding equation has a particular solution (1, 1, 1) and by Theorem 1 its general solution is

(−2x + y, y, z) = \frac{p}{q} · (m^{2} + 2mn − n^{2}, −m^{2} + 2mn + n^{2}, m^{2} + n^{2})

where (p, q) = 1 and q > 0 divides 4.

From this solution we express x = \frac{p}{q} · (−m^{2} + n^{2}) and y = \frac{p}{q} · (−m^{2} + 2mn + n^{2}) and plug them into the equation P_{2}(x, y) = g to obtain the following quartic equations:

m^{4} + 4m^{3}n − 6m^{2}n^{2} − 4mn^{3} + n^{4} = q^{2} · \frac{−1}{p^{2}}

and conclude that p^{2} = 1. Since the polynomial in the left hand side is irreducible, these are Thue equations.

We used PARI/GP to solve the resulting three Thue equations (for q = 1, 2, 4) and found that only the one with q = 2 has solutions, which are (m, n) = ±(5, −1), ±(1, 5), (±1, ±1) (the same solutions were earlier obtained by Lettl and Pethő [17]). The corresponding solutions to \frac{P_{1}(m,n)}{P_{2}(m,n)} = x^{2} are (m, n) = ±(12, 17) and ±(0, 1), giving respectively x = ±13 and ±1. Thus the solutions to Ljunggren equation are (±13, ±239) and (±1, ±1).

4 Near-multiples of squares in Lucas sequences

The pair of 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:

\begin{matrix}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, Q), n ≥ 1; \\ 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, Q), n ≥ 1. \end{matrix}

Some Lucas sequences have their own names:

Sequence Name  Initial terms  Index in [30]
U(1, −1) Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … \underline{A000045}
V (1, −1) Lucas numbers  2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, … \underline{A000032}
U(2, −1) Pell numbers  0, 1, 2, 5, 12, 29, 70, 169, 408, 985, … \underline{A000129}
V (2, −1) Pell-Lucas numbers 2, 2, 6, 14, 34, 82, 198, 478, 1154, … \underline{A002203}

Other examples include the Jacobsthal numbers U(1, −2), Mersenne numbers U(3, 2) etc.

The characteristic polynomial of Lucas sequences U(P, Q) and V (P, Q) is λ^{2} − P λ + Q with the discriminant D = P^{2} − 4Q. For non-degenerate sequences, the discriminant D is a positive non-square integer.

Terms of Lucas sequences satisfy the following identity:

V_{n}(P, Q)^{2} − D · U_{n}(P, Q)^{2} = 4Q^{n}.  (11)

In the current paper, we focus on the case of Q = 1 or Q = −1, implying that the pairs (V_{n}(P, Q), U_{n}(P, Q)) satisfy the equation:^{9}

x^{2} − Dy^{2} = ±4.  (12)

The converse statement can be used to prove that given positive integers belong to V (P, Q) or U(P, Q) respectively:

9Here and everywhere below ± in the r.h.s. of an equation means that we accept both signs as solutions.

Theorem 9 (Theorem 1 in [1]). Let P, Q be integers such that P > 0, |Q| = 1, (P, Q) ≠ (3, 1), and D = P^{2} − 4Q > 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.

Theorem 10. For fixed integers a ≠ 0 and b, finding terms of the form am^{2} + b in nondegenerate Lucas sequence U(P, Q) or V (P, Q) with Q = ±1 reduces to a finite number of Thue equations, unless this sequence is V (P, Q) and b = ±2.

Proof. Let ax^{2} + b be a term of U(P, ±1). By Theorem 9 we have y^{2} = D(ax^{2} + b)^{2} ± 4 = Da^{2}x^{4} + 2Dabx^{2} + (Db^{2} ± 4). Theorem 7 applies, since Db^{2} ± 4 ≠ 0 (notice that D ≠ ±1) and (2Dab)^{2} − 4Da^{2}(Db^{2} ± 4) = ∓16Da^{2} ≠ 0.

Now let ax^{2}+b be a term of V (P, ±1). By Theorem 9 we have (Dy)^{2} = D((ax^{2}+b)^{2}∓4) = Da^{2}x^{4} + 2Dabx^{2} + D(b^{2} ∓ 4). We have (2Dab)^{2} − 4D^{2}a^{2}(b^{2} ∓ 4) = ±16D^{2}a^{2} ≠ 0. Theorem 7 applies here as soon as b^{2} ∓ 4 ≠ 0 (i.e., b ≠ ±2).

Remark 11. For b = ±2, the sequence V (P, Q) may have an infinite number of terms of the form am^{2} + b. In particular, Lucas sequence V (1, −1) (Lucas numbers) has infinitely many terms of the forms m^{2} + 2 and m^{2} − 2 since V_{4n}(1, −1) = V_{2n}(1, −1)^{2} − 2 and V_{4n+2}(1, −1) = V_{2n+1}(1, −1)^{2} + 2.

The following theorem allows to find all terms solutions of the form am^{2} ± 2 in V (P, ±1).

Theorem 12. For fixed integers a ≠ 0 and b = ±2, finding terms of the form am^{2} + b in nondegenerate Lucas sequence V (P, Q) with Q = ±1 reduces to a finite number of Thue equations and a Pell-Fermat equation.

Proof. Let ax^{2} + b with b = ±2 be a term of V (P, ±1). By Theorem 9, we have the following equations:

(Dy)^{2} = D((ax^{2} + b)^{2} + 4) = Da^{2}x^{4} + 2Dabx^{2} + 8D

and

(Dy)^{2} = D((ax^{2} + b)^{2} − 4) = Da^{2}x^{4} + 2Dabx^{2}.

The former equation is addressed by Theorem 7, while the latter equation always has solution x = 0 (corresponding to the term V_{0} = 2) and for x ≠ 0 is equivalent to the Pell-Fermat equation:

(\frac{Dy}{x})^{2} − Da^{2}x^{2} = 2Dab.

For solution of this equation, we refer to [8, Section 6.3.5].

Theorem 10 implies finiteness of the terms of the form am^{2} + b in a Lucas sequence U(P, Q) or V (P, Q) with Q = ±1, unless this sequence is V (P, Q) and b = ±2. The latter case is addressed by Theorem 12 and may sometimes result in infinitely many terms. While this characterization represents a special case of the results by Nemes and Pethő [19] and by Pethő [20], our proofs rely on the simple transformation method developed in this paper.

In Table 1 we list all the terms of the form am^{2} + b for 1 ≤ a ≤ 3 and −3 ≤ b ≤ 3 in Fibonacci, Lucas, Pell, and Pell-Lucas numbers.

Fibonacci numbers Lucas numbers Pell numbers Pell-Lucas numbers
Form  U(1, −1)  V (1, −1)  U(2, −1)  V (2, −1)
m^{2}  0, 1, 144  1, 4  0, 1, 169  none
m^{2} + 1 1, 2, 5  1, 2  1, 2, 5  2, 82
m^{2} − 1 0, 3, 8  3  0  none
m^{2} + 2 2, 3  2, 11, and V_{4n+2} 2  2 and V_{4n+2}
m^{2} − 2 2, 34  V_{4n}  2  14 and V_{4n}
m^{2} + 3 3  3, 4, 7, 199  12  none
m^{2} − 3 1, 13, 1597  1  1  6
2m^{2}  0, 2, 8  2, 18  0, 2  2
2m^{2} + 1 1, 3  1, 3  1  none
2m^{2} − 1 1  1, 7, 199  1  none
2m^{2} + 2 2, 34  2, 4  2  2 and V_{4n}
2m^{2} − 2 0  none  0, 70  V_{4n+2}
2m^{2} + 3 3, 5, 21  3, 11  5  none
2m^{2} − 3 5  29, 47, 64079  5, 29  none
3m^{2}  0, 3  3  0, 12  none
3m^{2} + 1 1, 13  1, 4, 76  1  none
3m^{2} − 1 2  2, 11, 47  2  2
3m^{2} + 2 2, 5  2, 29  2, 5, 29  2, 14
3m^{2} − 2 1  1  1  none
3m^{2} + 3 3  3  none  6
3m^{2} − 3 0, 144  none  0  none

Table 1: Terms of the form am^{2} + b for 1 ≤ a ≤ 3 and −3 ≤ b ≤ 3 in Fibonacci, Lucas, Pell, and Pell-Lucas numbers.

5 Acknowledgements

We thank Dr. John Cremona, Dr. Nikos Tzanakis, and Dr. Noam Elkies for a number of helpful suggestions and discussions.

The first author was supported by the National Science Foundation under Grant No. IIS-1253614. The second author was partially supported by the European Union and the European Social Fund through project “Supercomputer, the national virtual lab” under Grant No. TÁMOP-4.2.2.C-11/1/KONV-2012-0010.

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2010 Mathematics Subject Classification: Primary 11Y50; Secondary 11D25, 11B39, 14G05. Keywords: integral point, biquadratic curve, elliptic curve, Thue equation, Fibonacci number, Lucas sequence.

(Concerned with sequences \underline{A000032}, \underline{A000045}, \underline{A000129}, \underline{A002203}.)

Received February 22 2014; revised version received May 17 2014. Published in Journal of Integer Sequences, May 18 2014.

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