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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, $1913$.

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What does "230" mean? –  Tom Leinster Feb 9 '11 at 8:50
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Number of the problem, I guess. –  darij grinberg Feb 9 '11 at 9:25
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I thought that Mr. Scott is offering 230 USD for this ;) –  Piotr Achinger Feb 9 '11 at 10:18
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Move along, there's nothing to see here. –  Franz Lemmermeyer Feb 9 '11 at 10:42
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@Luis: the faq has a couple of suggestions concerning how to ask a question and how to choose a title. Is your question the one awllower asked, or did you have something else in mind? –  Franz Lemmermeyer Feb 9 '11 at 13:10
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1 Answer

up vote 12 down vote accepted

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.)

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