Four Biquadrates Whose Sum is a Perfect Square

Ajai Choudhry 13/4 A Clay Square Lucknow 226001 India

                  ajaic203@yahoo.com

Abstract

All known nontrivial parametric solutions of the diophantine equation

x_{1}^{4} + x_{2}^{4} + x_{3}^{4} + x_{4}^{4} = y^{2}

give the values of x_{i}, i = 1, …, 4, in terms of polynomials in two parameters. In the three simplest solutions, these polynomials are of degrees 2, 4 and 12 respectively. In this paper we obtain infinitely many solutions of the equation under consideration in terms of quadratic polynomials in two parameters, as well as a solution in terms of quartic polynomials in three parameters. We also show how more general multiparameter solutions of the equation may be obtained.

1 Introduction

This paper is concerned with the problem of finding four biquadrates whose sum is a perfect square, that is, we are required to find four integers x_{i}, i = 1, …, 4, such that

x_{1}^{4} + x_{2}^{4} + x_{3}^{4} + x_{4}^{4} = y^{2}.  (1)

where y is an integer. If we take three of the integers x_{i}, i = 1, …, 4, as 0, we immediately get a trivial solution of Eq. (1). All other solutions of the diophantine equation (1) will be considered as nontrivial.

All known nontrivial parametric solutions of Eq. (1) are expressed in terms of polynomials in two parameters—the simplest being given by the identity,

(a^{4} + b^{4})^{2} = a^{8} + (ab)^{4} + (ab)^{4} + b^{8}.

Dickson [3, pp. 657–658]) has mentioned two other solutions—found by Fauquembergue—in which the values of x_{i}, i = 1, …, 4, are given by polynomials of degrees 4 and 12 in terms of two parameters. Recently Alvarado and Delorme [1] have obtained two solutions in which x_{i}, i = 1, …, 4, are given by polynomials of degrees 16 and 28, and they have shown how infinitely many such solutions may be obtained by using elliptic curves.

In this paper we will obtain infinitely many solutions of Eq. (1) in terms of quadratic polynomials in two parameters, as well as a solution in terms of quartic polynomials in three parameters. We also show how more general multi-parameter solutions of the equation may be obtained.

2 Four biquadrates whose sum is a square

2.1 Solutions in terms of quadratic polynomials in two parameters

In this section we will obtain solutions of the diophantine equation (1) in terms of polynomials of degree 2 in two parameters by using the well-known composition of forms identity,

(f^{2} + fg + g^{2})(u^{2} + uv + v^{2}) = p^{2} + pq + q^{2},  (2)

where f, g, u, v are arbitrary parameters and

p = fu − gv, q = fv + gu + gv,

together with the identity,

r^{4} + s^{4} + (r + s)^{4} = 2(r^{2} + rs + s^{2})^{2},  (3)

where r and s are arbitrary parameters.

It follows from (2) that

\begin{aligned}(f^{2} + fg + g^{2})(u^{2} + uv + v^{2})^{2} & = (p^{2} + pq + q^{2})(u^{2} + uv + v^{2}), \\ & = r^{2} + rs + s^{2}, \end{aligned}  (4)

where

r = fu^{2} − 2guv − (f + g)v^{2}, s = gu^{2} + (2f + 2g)uv + fv^{2}.  (5)

Thus, when the values of r and s are given by (5), we get, on combining the two identities (3) and (4),

r^{4} + s^{4} + (r + s)^{4} = 2(f^{2} + fg + g^{2})^{2}(u^{2} + uv + v^{2})^{4},  (6)

and on adding h^{4}(u^{2} + uv + v^{2})^{4} to both sides of (6), we get the identity,

r^{4} + s^{4} + (r + s)^{4} + h^{4}(u^{2} + uv + v^{2})^{4} = (2(f^{2} + fg + g^{2})^{2} + h^{4})(u^{2} + uv + v^{2})^{4}. (7)

If we choose integer values of f, g, and h such that

2(f^{2} + fg + g^{2})^{2} + h^{4} = t^{2},  (8)

where t is some integer, it follows from (5) and (7) that a solution of (1) is given by

\begin{aligned}(x_{1}, x_{2}, x_{3}, x_{4}, y) & = (fu^{2} − 2guv − (f + g)v^{2}, gu^{2} + (2f + 2g)uv + fv^{2}, \\ & (f + g)u^{2} + 2fuv − gv^{2}, h(u^{2} + uv + v^{2}), t(u^{2} + uv + v^{2})^{2}), \end{aligned}

where u and v are arbitrary parameters.

We readily found, by computer trials, the following four sets of values of (f, g, h, t) that satisfy Eq. (8):

(2, 2, 1, 17), (2, 4, 7, 63), (2, 14, 17, 433), (4, 22, 7, 833).

These four solutions of Eq. (8) immediately yield four solutions of Eq. (1) in which the values of x_{i} are given by quadratic polynomials in the arbitrary parameters u and v. The first two solutions of Eq. (1) thus obtained may be written explicitly as follows:

\begin{aligned}(x_{1}, x_{2}, x_{3}, x_{4}, y) & = (2u^{2} − 4uv − 4v^{2}, 2u^{2} + 8uv + 2v^{2}, 4u^{2} + 4uv − 2v^{2}, \\ & u^{2} + uv + v^{2}, 17(u^{2} + uv + v^{2})^{2}), \end{aligned}  (9)

and

\begin{aligned}(x_{1}, x_{2}, x_{3}, x_{4}, y) & = (2u^{2} − 8uv − 6v^{2}, 4u^{2} + 12uv + 2v^{2}, 6u^{2} + 4uv − 4v^{2}, \\ & 7(u^{2} + uv + v^{2}), 63(u^{2} + uv + v^{2})^{2}). \end{aligned}  (10)

We will now show how infinitely many solutions in integers of Eq. (8) may be obtained. For fixed rational numerical values of g and h, Eq. (8) is a quartic model of an elliptic curve. If this elliptic curve is of positive rank, we can obtain infinitely many rational points on this elliptic curve, and by appropriate scaling, we can obtain infinitely many integer solutions of Eq. (8). For instance, when (g, h) = (2, 1), we may write Eq. (8) as

2f^{4} + 8f^{3} + 24f^{2} + 32f + 33 = t^{2},  (11)

and one rational point on the curve (11) is known to be (f, t) = (2, 17).

The birational transformation defined by

\begin{aligned}f & = −2(145X + 17Y + 1208)/(144X − 17Y + 1176), \\ t & = (4913X^{3} + 121584X^{2} + 982600X − 6864Y + 2585088) \\ & × (144X − 17Y + 1176)^{−2}, \end{aligned}  (12)

and

\begin{aligned}X & = 2(20f^{2} + 64f + 17t + 81)/(f − 2)^{2}, \\ Y & = 4(102f^{3} + 408f^{2} + 72ft + 816f + 145t + 833)/(f − 2)^{3}, \end{aligned}  (13)

reduces the elliptic curve (11) to the Weierstrass form,

Y^{2} = X^{3} − 200X − 1088.  (14)

A reference to Cremona’s elliptic curve tables [2] shows that the rank of the elliptic curve (14) is 1. In fact, using APECS (a package written in Maple for working with elliptic curves), we readily found the rational point P = (217/9, 2431/27) on the curve (14). Since the point P does not have rational coordinates, it follows from the Nagell-Lutz theorem [4, p. 56] on elliptic curves that P is a point of infinite order. We can now find infinitely many rational points on the curve (14) using the group law, and using the relations (12), we can find infinitely many rational points on the curve (11). We can now obtain infinitely many integer solutions of Eq. (8) and thus generate infinitely many parametric solutions of Eq. (1).

As a numerical example, the point P on the curve (14) generates a solution that is equivalent to the solution (9) while the point 2P generates the following parametric solution of Eq. (1):

\begin{aligned}x_{1} & = 162278u^{2} + 667292uv + 171368v^{2}, \\ x_{2} & = 333646u^{2} + 342736uv − 162278v^{2}, \\ x_{3} & = 171368u^{2} − 324556uv − 333646v^{2}, \\ x_{4} & = 166823(u^{2} + uv + v^{2}), \\ y & = 121336214273(u^{2} + uv + v^{2})^{2}, \end{aligned}

where u and v are arbitrary parameters.

2.2 Solutions in terms of quartic polynomials in three parameters

We will now obtain a solution of (1) in which the values of x_{i}, i = 1, …, 4, are given by quartic polynomials in three parameters.

We write

x_{1} = t + a, x_{2} = t + b, x_{3} = t + c, x_{4} = t,  (15)

when the left-hand side of Eq. (1) becomes a quartic function of t in which the coefficient of t^{4} is a perfect square, namely 4. Dickson [3, p. 639] has described a method, originally given by Fermat, for making such a quartic function a perfect square. Applying this method, we find that the left-hand side of Eq. (1) becomes a perfect square if we take

t = ϕ(a, b, c)/(24(a + b − c)(a − b + c)(a − b − c)),

where

\begin{aligned}ϕ(a, b, c) & = 9(a^{4} + b^{4} + c^{4}) − 20(a^{3}(b + c) + b^{3}(c + a) + c^{3}(a + b)) \\ & + 54(a^{2}b^{2} + b^{2}c^{2} + c^{2}a^{2}) − 12abc(a + b + c). \end{aligned}

Using the relations (15), we now get four biquadrates whose sum is a perfect square. On appropriate scaling, we may write these biquadrates as x_{i}^{4}, i = 1, …, 4, with the values of x_{i} being given by the quartic polynomials,

x_{1} = ψ(a, b, c), x_{2} = ψ(b, c, a), x_{3} = ψ(c, a, b), x_{4} = ϕ(a, b, c),  (16)

where

ψ(a, b, c) = 33a^{4} − 44a^{3}b − 44a^{3}c + 30a^{2}b^{2} + 36a^{2}bc
+ 30a^{2}c^{2} + 4ab^{3} − 36ab^{2}c − 36abc^{2} + 4ac^{3}
+ 9b^{4} − 20b^{3}c + 54b^{2}c^{2} − 20bc^{3} + 9c^{4}.

On making the invertible linear transformation,

a = (v + w)/2, b = (u + w)/2, c = (u + v)/2,

the values of x_{i} given by (16) may be written as

\begin{aligned}x_{1} & = f(u, −v, −w), & x_{2} & = (−u, v, −w), \\ x_{3} & = f(−u, −v, w), & x_{4} & = f(u, v, w), \end{aligned}  (17)

where f(u, v, w) is defined in terms of arbitrary parameters u, v, and w as follows:

f(u, v, w) = 2(u^{4} + v^{4} + w^{4}) + 3(u^{2}v^{2} + v^{2}w^{2} + w^{2}u^{2}) + 6uvw(u + v + w).

With the values of x_{i}, i = 1, …, 4, given by (17), it is readily verified that ∑_{i=1}^{4} x_{i}^{4} = y^{2} where

\begin{aligned}y & = 8(u^{8} + v^{8} + w^{8}) + 24(u^{6}(v^{2} + w^{2}) + v^{6}(w^{2} + u^{2}) + w^{6}(u^{2} + v^{2})) \\ & + 34(u^{4}v^{4} + v^{4}w^{4} + w^{4}u^{4}) + 276u^{2}v^{2}w^{2}(u^{2} + v^{2} + w^{2}). \end{aligned}  (18)

Thus a solution of Eq. (1) is given by (17) and (18) in terms of the arbitrary parameters u, v, and w.

2.3 Multi-parameter solutions

We will now describe a method of obtaining more general solutions of Eq. (1) in several parameters by using the known solutions of Eq. (1).

Let a solution of Eq. (1) be (x_{1}, x_{2}, x_{3}, x_{4}, y) = (α_{1}, α_{2}, α_{3}, α_{4}, β). Instead of choosing the values of x_{i}, i = 1, …, 4, given by (15), we now write,

\begin{aligned}x_{1} & = α_{1}t_{1} + γ_{1}t_{2}, x_{2} & = α_{2}t_{1} + γ_{2}t_{2}, \\ x_{3} & = α_{3}t_{1} + γ_{3}t_{2}, x_{4} & = α_{4}t_{1} + γ_{4}t_{2}, \end{aligned}  (19)

where γ_{i}, i = 1, …, 4, are arbitrary parameters, and proceeding as before, we again observe that the left-hand side of Eq. (1) becomes a quartic function of t_{1} in which the coefficient of t_{1}^{4} is a perfect square, namely β^{2} and hence, as before, we can obtain a value of t_{1} that yields four biquadrates whose sum is a perfect square. We thus obtain more parametric solutions of Eq. (1).

We could also choose the parameters γ_{i} in (19) such the sum of the four biquadrates γ_{i}^{4} is a perfect square, say δ^{2}, and now the left-hand side of Eq. (1) becomes a binary quartic form in t_{1} and t_{2} such that the coefficients of t_{1}^{4} and t_{2}^{4} are β^{2} and δ^{2}, respectively. If we now write

y = βt_{1}^{2} + 2 (∑_{i=1}^{4} α_{i}^{3}γ_{i}) t_{1}t_{2}/β + δt_{2}^{2},  (20)

and substitute the values of x_{i}, i = 1, …, 4, and y given by (19) and (20), respectively, in Eq. (1), we get after suitable transposition and removal of common factors, a linear equation in t_{1} and t_{2} whose solution is as follows:

\begin{aligned}t_{1} & = 2 (∑_{i=1}^{4} α_{i}γ_{i}^{3}) β^{2} − 2 (∑_{i=1}^{4} α_{i}^{3}γ_{i}) βδ, \\ t_{2} & = β^{3}δ − 3 (∑_{i=1}^{4} α_{i}^{2}γ_{i}^{2}) β^{2} + 2 (∑_{i=1}^{4} α_{i}^{3}γ_{i})^{2} . \end{aligned}  (21)

With the values of t_{1} and t_{2} given by (21), a solution of Eq. (1) is given by (19) and (20). Since we can choose the values of α_{i}, γ_{i}, i = 1, …, 4, in terms of 2 or 3 parameters, the solutions of Eq. (1) thus obtained are in terms of 4, 5 or 6 parameters. All the multiparameter solutions of Eq. (1) obtained in this manner are too cumbersome to write and accordingly we do not give any of them explicitly.

It would of considerable interest to find a parametric solution of Eq. (1) in which the parameters can be so chosen that the value of y becomes a perfect square. If that can be achieved, we will obtain four biquadrates whose sum is also a biquadrate. There seems to be no simple way of finding such a parametric solution of Eq. (1).

3 Acknowledgments

I wish to thank the Harish-Chandra Research Institute, Prayagraj for providing me with all necessary facilities that have helped me to pursue my research work in mathematics.

References

[1] A. Alvarado and J.-J. Delorme, On the diophantine equation x^{4} + y^{4} + z^{4} + t^{4} = w^{2}, J. Integer Sequences 17 (2014), Article 14.11.5.

[2] J. E. Cremona, Elliptic curve data. Available at http://johncremona.github.io/

   ecdata/.

[3] L. E. Dickson, History of the Theory of Numbers, Vol. 2, Chelsea Publishing Company, 1952.

[4] J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Springer-Verlag, 1992.

2010 Mathematics Subject Classification: 11D25. Keywords: biquadrate; quartic diophantine equation.

Received October 9 2020; revised version received January 18 2021. Published in Journal of Integer Sequences, January 23 2021.

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