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The Diophantine equation \(xy+yz+xz=n\) and indecomposable binary quadratic forms. (English) Zbl 1147.11314

It is well-known that there are 18 (and possibly 19) integers that are not of the form \( xy + yz + xz \) with positive integers \(x, y, z\) [see F. Z. Zhu and Y. Y. Shao, Chin. Ann. Math., Ser. B 9, No. 1, 79–94 (1988; Zbl 0658.10027); M. Peters, Arch. Math. 57, No. 5, 467–468 (1991; Zbl 0707.11025)]. The same 18 integers appear as exceptional discriminants for which no indecomposable positive definite binary quadratic form exists [J. M. Borwein and K.-K. S. Choi, Exp. Math. 9, No. 1, 153–158 (2000; Zbl 0970.11011)]. Here the author shows that the two problems are equivalent.

MSC:

11E12 Quadratic forms over global rings and fields
11D09 Quadratic and bilinear Diophantine equations
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References:

[1] Borwein J. M., Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity (1987)
[2] Borwein J., Exp. Math. 9 (1) pp 153– (2000) · Zbl 0970.11011
[3] Le Maohua, Publ. Math. Debrecen 52 pp 159– (1998)
[4] O’Meara O. T., Introduction to Quadratic Forms. (1963)
[5] DOI: 10.1007/BF01246744 · Zbl 0707.11025
[6] Zhu F. Z., Chinese Ann. Math. Ser. 13(1) 9 pp 79– (1988)
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