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Solving \(n(n + d) \cdots (n + (k - 1)d) = by^{2}\) with \(P(b) \leq Ck\). (English) Zbl 1268.11045

The equation in the title has a long history. Under the usual assumption \(P(b)\leq k\) there are several effective and numerical results in the literature. Here \(P(t)\) stands for the greatest prime divisor of a positive integer \(t\), with the convention \(P(1)=1\). Under some mild restrictions, the authors prove that for any fixed \(C\) the equation has only finitely many solutions, and these can be effectively computed. To handle the problem in this generality, the authors introduce some new nice and interesting ideas of combinatorial nature.

MSC:

11D45 Counting solutions of Diophantine equations
11D72 Diophantine equations in many variables
11B25 Arithmetic progressions
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References:

[1] DOI: 10.1017/CBO9780511565977 · doi:10.1017/CBO9780511565977
[2] DOI: 10.1112/S0024611505015625 · Zbl 1178.11033 · doi:10.1112/S0024611505015625
[3] DOI: 10.1006/jnth.1996.0129 · Zbl 0867.11017 · doi:10.1006/jnth.1996.0129
[4] DOI: 10.1007/978-1-4757-5927-3 · doi:10.1007/978-1-4757-5927-3
[5] Erdos P., Nieuw Arch. Wisk. (3) 3 pp 124–
[6] Erdos P., Illinois J. Math. 19 pp 292–
[7] Filaseta M., J. London Math. Soc. 45 pp 215–
[8] DOI: 10.1112/S0010437X09004114 · Zbl 1194.11043 · doi:10.1112/S0010437X09004114
[9] Laishram S., Pub. Math. Debrecen 68 pp 451–
[10] DOI: 10.1016/S0019-3577(06)80042-X · Zbl 1165.11014 · doi:10.1016/S0019-3577(06)80042-X
[11] Lehmer D. H., Illinois J. Math. 8 pp 57–
[12] DOI: 10.1007/BF01203608 · JFM 46.0240.04 · doi:10.1007/BF01203608
[13] DOI: 10.1090/S0025-5718-96-00669-2 · Zbl 0856.11042 · doi:10.1090/S0025-5718-96-00669-2
[14] Rosser J. B., Illinois J. Math. 6 pp 64–
[15] DOI: 10.1112/S0010437X04001125 · Zbl 1167.11305 · doi:10.1112/S0010437X04001125
[16] T. N. Shorey, A Panorama of Number Theory or the View from Baker’s Garden (Cambridge University Press, Cambridge, 2002) pp. 325–336.
[17] Shorey T. N., Acta Arith. 61 pp 391–
[18] DOI: 10.1016/0022-314X(89)90014-0 · Zbl 0657.10014 · doi:10.1016/0022-314X(89)90014-0
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