A New Solution to a Cubic Diophantine Equation

: A positive integer, which can be written as the sum of two positive cubes in two different ways, is known as a “Ramanujan number”. The most famous example is 1729 = 10 3 + 9 3 = 12 3 + 1 3 , which was identiﬁed by Ramanujan as the lowest such number. In this paper, we consider the homogeneous cubic Diophantine equation x 3 + y 3 = u 3 + v 3 , where there is no restriction on the signs of the integers x , y , u , v . We show that every solution can be written in terms of two parameters in the ring Z (cid:0) √− 3 (cid:1) . It is also shown that solutions with arbitrarily high values of max ( | x | , | y | , | u | , | v | ) arise amongst the primitive solutions.


Introduction
The aim of this paper is to give a new solution to the Diophantine equation x, y, u, v ∈ Z.
The result found in this paper gives the general solution in terms of two parameters in the ring R := Z √ −3 . Depending on the signs of the integers in any particular solution of (1), we obtain a solution to one or the other of x, y, u, v ∈ Z + , (2) x, y, z, w ∈ Z + .
In Section 2, we consider some properties of R, and in Section 3, we present a solution to (1). This is followed by Section 3, where the cubic Diophantine Equation (3) is considered, and this leads to an algorithm for generating all possible solutions in Section 4. Finally, in Section 5, we show how to construct primitive solutions with arbitrarily large values.

The Ring
−3, then ξη = (ac − 3bd) + (ad + bc) √ −3. We consider the question: "which odd primes greater than 3 are equal to ξ for some ξ"? Lemma 1. Let p be a prime greater than 3; then, there exists a + b √ −3 ∈ R with p a, p b, such that a 2 + 3b 2 ≡ 0 mod p, if and only if p ≡ 1 mod 6.
Proof. Choose the integer c such that bc = 1 mod p and multiply (4) by c and write d = ac. It follows that −3 ≡ d 2 mod p, implying that, using Legendre symbols, Denote P as the set of primes that are congruent to 1 mod 6. For a given p ∈ P, denote Lemma 2. Let p ∈ P; then, there exist integers a, b ∈ N such that Proof. Replace a by a in (5), where a = np ± a with a ∈ N and similarly for b.
Theorem 1. Let p ∈ P; then, a unique |a| and |b| exist such that Proof. The proof is by induction for p = 7, 13, 19, · · · ∈ P. For p = 7, it is verified that 2 2 + 3(1) 2 = p. It remains for us to prove the result for p ∈ P, p > 7, where it is assumed to hold for all q ∈ P, such that q < p, From Lemma 2, a, b exist such that a 2 + 3b 2 = pQ, with Q a product of prime powers. We show that this statement is also true with Q = 1. It is not possible that p | Q because The proof continues by considering a series of propositions that make it possible to exclude all possible prime divisors of Q. In each case, for a possible divisor g, we show that there exist a , b such that a 2 + 3b 2 = pQ with Q < Q. In this proof only, the symbol ⇒ ⇒ ⇒ is used to introduce values of a , b , Q . i. 4. q | Q, for q prime and q ≡ 5 mod 6. It follows that case 1 applies with g = q.
(a) q | a, q | b. Reduces to case 1.
We now consider the primes of R and factorization in R.
Remark 2. The product of two primes of R has a unique factorization with the exception of

Solution to Cubic Equation
We first state the solution to an introductory equation.

Lemma 3. The general solution to
Consider two Diophantine equations a(a 2 + 3b 2 ) = c(c 2 + 3d 2 ) In each case, we consider primitive solutions. The next result is easily verified and is given without proof.
We now focus on (10), but written in a slightly different form.
We use Lemma 3 to solve (11) and then identify the free parameters by requiring them to satisfy (12).

Discussion 1.
To eliminate r, s, t, use the conditions a = a, c = c to obtain so that r, s, t satisfy the homogeneous linear system where G −1 is introduced to reduce gcd(r, s, t) to 1, if necessary.

Algorithm and Sample Solutions
Using the results of Lemmas 4, 5 and Discussion 1, an algorithm can be constructed for finding solutions to Evaluate r, s, t; 3.
Evaluate a, b, c, d;
It is found computationally that there are exactly 25 solutions to (2) with x 3 + y 3 ≤ 1000, 000 and 31 solutions to (3) with w ≤ 100. These are shown in Table 1.

Conclusions
A new parametric solution to the homogeneous cubic equation x, y, u, v ∈ Z, is derived using a pair of parameters in the ring Z( √ −3). As a consequence, it is shown that, amongst solutions of (26), there exist arbitrarily large values of N = x 3 + y 3 , x, y, u, v > 0. Furthermore, it is shown that, amongst solutions of x 3 + y 3 + z 3 = w 3 , x, y, z, w > 0, there exist arbitrarily large values of w.