Abstract
Let \(f(x, k, d) = x(x + d)\cdots(x + (k - 1)d)\) be a polynomial with \(k \geq 2, d \geq 1\). We consider the Diophantine equation \(\prod_{i=1}^{r} f(x_i, k_i, d) = y^{2}, r \geq 1\). Using the theory of Pell equations, we affirm a conjecture of Bennett and van Luijk [3]; extend some results of this Diophantine equation for \(d=1\), and give a positive answer to Question 3.2 of Zhang [19].
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Bauer, M., Bennett, M.A.: On a question of Erdős and Graham. Enseign. Math. 53, 259–264, 2007
Bennett, M.A., Bruin, N., Győry, K., Hajdu, L.: Powers from products of consecutive terms in arithmetic progression. Proc. Lond. Math. Soc. 92, 273–306, 2006
M. A. Bennett and R. Van Luijk, Squares from blocks of consecutive integers: a problem of Erdős and Graham, Indag. Math. (N.S.), 23 (2012), 123–127.
P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, Monograph Enseign. Math., 28, Université de Genève (Geneva, 1980).
Erdős, P., Selfridge, J.L.: The product of consecutive integers is never a power. Illinois J. Math. 19, 292–301, 1975
Guy, R.K.: Unsolved Problems in Number Theory, 3rd edn. Springer-Verlag (Berlin 2004).
Győry, K., Hajdu, L., Saradha, N.: On the Diophantine equation \(n(n+d)\cdots (n+(k-1)d)=by^l\). Canad. Math. Bull. 47, 373–388, 2004
L. Hajdu, Sz. Tengely and R. Tijdeman, Cubes in products of terms in arithmetic progression, Publ. Math. Debrecen, 74 (2009), 215–232.
Hirata-Kohno, N., Laishram, S., Shorey, T.N., Tijdeman, R.: An extension of a theorem of Euler. Acta Arith. 129, 71–102, 2007
Katayama, S.: Products of arithmetic progressions which are squares. J. Math. Tokushima Univ. 49, 9–12, 2015
Luca, F., Walsh, P.G.: On a Diophantine equation related to a conjecture of Erdős and Graham. Glas. Mat. Ser. III(42), 281–289, 2007
Obláth, R.: Über das Produkt fünf aufeinander folgender Zahlen in einer arithmetischen Reihe. Publ. Math. Debrecen 1, 222–226, 1950
O. Rigge, Über ein diophantisches Problem, in: 9th Congress Math. Scand. (Helsingfors, 1938), Mercator (1939), pp. 155–160.
Skałba, M.: Products of disjoint blocks of consecutive integers which are powers. Colloq. Math. 98, 1–3, 2003
Sz. Tengely, On a problem of Erdős and Graham, Period. Math. Hungar., 72 (2016), 23–28.
Sz. Tengely and M. Ulas, On products of disjoint blocks of arithmetic progressions and related equations, J. Number Theory, 165 (2016), 67–83.
Ulas, M.: On products of disjoint blocks of consecutive integers. Enseign. Math. 51, 331–334, 2005
B. Yıldız and E. Gürel, On a problem of Erdős and Graham, Bull. Braz. Math. Soc., New Series, 51 (2020), 397–415.
Y. Zhang, On products of consecutive arithmetic progressions. II, Acta Math. Hungar., 156 (2018), 240–254.
Zhang, Y., Cai, T.: On products of consecutive arithmetic progressions. J. Number Theory 147, 287–299, 2015
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This research was supported by the National Natural Science Foundation of China (Grant No. 11501052), Younger Teacher Development Program of Changsha University of Science and Technology (Grant No. 2019QJCZ051), Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science and Technology) and the Natural Science Foundation of Zhejiang Province (Project No. LY18A010016).
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Zhang, Y. On products of consecutive arithmetic progressions. III. Acta Math. Hungar. 163, 407–428 (2021). https://doi.org/10.1007/s10474-020-01108-4
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DOI: https://doi.org/10.1007/s10474-020-01108-4