On the equationa 3+b 3+c 3=d 3

Binary quadratic forms and sums of powersof integers

“…Regarding formula (3.4), it looks somewhat more exotic, although it is by no means new. An equivalent formulation of both Equations (3.3) and (3.4) can be found in, respectively, formulas (17) and (22) of the review paper by Kotiah [14]. It is worth pointing out, on the other hand, that the right-hand side of Equation (3 Equation (3.6) can in turn be written explicitly as a function of the variable by expressing each of the involved power sums in terms of .…”

“…As noted by Sándor, relations (4) can be obtained by generalizing Ramanujan's quadratic solution (2) but, following [22], we will give a simpler proof of Theorem 1 employing a technique devised by Nicholson [18]. Thus, let a 3 + b 3 + c 3 = d 3 be a nontrivial solution of (1), and consider Nicholson's parametric equation [22] ux − cy…”

“…(1) Substituting (a, b, c, d) = (3,4,5,6) into Equations ( 4), dividing by 2, and using the linear transformation u → u/2, we get Ramanujan's solution (2) [22]. (2) Using (a, b, c, d) = (1,6,8,9) into Equations ( 4) and dividing by 3 yields…”

“…As was anticipated in the introduction, we shall make use of a theorem of Sándor (see [22,Theorem 1]) in order to construct two-parameter quadratic solutions for the cubic equation (1). Following Sándor, we say that a solution of (1) is trivial if d = a or d = b or d = c. The said theorem, adapted to our notation, is as follows.…”

“…Our interest in this type of solutions stems from the fact that, as explained in [12], by using the above identity 3 3 + 4 3 + 5 3 = 6 3 as a seed, one can generate quadratic-form formulas to Equation (1). Expanding on this point, and borrowing a theorem of Sándor [22,Theorem 1], in Section 2 we show that, indeed, it is possible to construct quadraticform representations for the cubic equation (1) starting from any particular nontrivial solution to (1) (see below for the definition of a trivial solution). The proof of this result given by Sándor (which is essentially reproduced in Section 2) is particularly suitable for our purpose since it utilizes only precalculus tools.…”