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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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    $\begingroup$ What does "230" mean? $\endgroup$
    – Tom Leinster
    Commented Feb 9, 2011 at 8:50
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    $\begingroup$ I thought that Mr. Scott is offering 230 USD for this ;) $\endgroup$
    – Piotr Achinger
    Commented Feb 9, 2011 at 10:18
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    $\begingroup$ Move along, there's nothing to see here. $\endgroup$
    – Franz Lemmermeyer
    Commented Feb 9, 2011 at 10:42
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    $\begingroup$ @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? $\endgroup$
    – Franz Lemmermeyer
    Commented Feb 9, 2011 at 13:10
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    $\begingroup$ The problem was proposed by E. B. Escott (not Scott), possibly the Escott of Tarry-Escott, q.v. $\endgroup$
    – Gerry Myerson
    Commented Feb 9, 2011 at 23:30

1 Answer 1

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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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