We give a complete description of all solutions to the equation f13 + f 23 = f 33 + f 43 for quadratic forms fj ∈ ℂ[x,y] and show how Ramanujan’s example can be extended to three equal sums of pairs of cubes. We also give a complete census in counting the number of ways a sextic p ∈ ℂ[x,y] can be written as a sum of two cubes. The extreme example is p(x,y) = xy(x4 − y4), which has six such representations.

中文翻译：

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二次型的两个立方的等和

我们给出方程的所有解的完整描述F13 + F 23 = F 33 + F 43对于二次形式Fj ∈ ℂ[X,是的]并展示如何将 Ramanujan 的示例扩展到三个相等的立方体对的总和。我们还提供了一个完整的人口普查，计算了六分法的数量p ∈ ℂ[X,是的]可以写成两个立方体的和。极端的例子是p(X,是的) = X是的(X4 - 是的4)，它有六个这样的表示。