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A Diophantine equation. (English) Zbl 0576.10010
The author finds by the elementary theory of algebraic number fields all the cases when the sum of three consecutive integral cubes is a square; that is to find all integral solutions \(x,y\) of \(y^ 2=(x-1)^ 3+x^ 3+(x+1)^ 3=3x(x^ 2+2)\).
Editor’s remark: In an addendum in Glasg. Math. J. 29, 275 (1987; Zbl 0642.10019) priority of this result for S. Uchiyama is acknowledged.

MSC:
11D25 Cubic and quartic Diophantine equations
14G20 Local ground fields in algebraic geometry
11D88 \(p\)-adic and power series fields
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[1] DOI: 10.1112/plms/s3-14A.1.55 · Zbl 0134.27501
[2] DOI: 10.1093/qmath/20.1.129 · Zbl 0177.06802
[3] DOI: 10.1017/S0305004100045904
[4] Baker, Transcendental number theory (1975)
[5] DOI: 10.1016/0022-314X(72)90058-3 · Zbl 0236.10010
[6] Mordell, Diophantine equations (1969)
[7] Ljunggren, Avh. Norske Vid.-Akad. Oslo 5 pp 1– (1942)
[8] DOI: 10.2307/3617190
[9] Skolem, Bull. Soc. Math. Belg. 7 pp 83– (1955)
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