Summary: In my paper [J. Lond. Math. Soc. 11, 208–218 (1936; Zbl 0014.20302)] I proved the result that if \(a, b, c, d\) are integers, there are an infinity of integer solutions of \(x^3+y^3+z^3+w^3 = a^3+b^3+c^3+d^3\) when any one of the three numbers \(-(a+b)(c+d)\), etc. is positive and not a perfect square. Prof. Sierpiński has kindly pointed out that the proof does not hold when \(a = b\), \(c = d\), etc. Hence it is not known if
\[x^3+y^3+z^3+w^3 = 2a^3+2b^3quad\text{has an infinity of integer solutions}. \]
In fact, the result stated at the end of page 216, that an infinity of solutions exist, is obviously subject to the further condition \(AC^2+BD^2 \ne 0\), i.e. \((a+b)(a-b)^2 + (c+d)(c-d)^2 \ne 0\). Since \(-AB\) is not a perfect square, this inequality is satisfied unless \(C = 0\), \(D = 0\), i.e. unless \(a = b\), \(c = d\).