$230.$ (April, 1915) Proposed by E. B. Escott, Ann Arbor, Michigan - MathOverflow

$230.$ (April, 1915) Proposed by E. B. Escott, Ann Arbor, Michigan

2011-02-09T08:41:48Z

Just browsing some old stuff in my office for other thing I found the following:

$230.$ (April, 1915) Proposed by E. B. Escott, Ann Arbor, Michigan.

Find three numbers such that their sum, the sum of their squares, and the sum of their cubes , shall be a cube.

Note.--W. D. Cairns says this problem, which was proposed in L'Intermediaire in $1900$, remains unsolved to date, even though it was reprinted in February, $1913$.

Answer by Tapio Rajala for $230.$ (April, 1915) Proposed by E. B. Escott, Ann Arbor, Michigan

2011-02-09T12:14:15Z

With a quick search from the internet I was able to only find the solution $$(146, -1314, 1168)$$ by E. T. Bell in The American Mathematical Monthly, Vol. 24, No. 5 (May, 1917), p. 240. The paper can be found from http://www.jstor.org/stable/pdfplus/2974328.pdf

(Also, a quick computer search shows that there are no positive solutions with the largest integer being less than 90000.)