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On products of consecutive arithmetic progressions. III. (English) Zbl 1480.11035

In the paper the authors are interested in integer solutions of the Diophantine equation \[ y^2=\prod_{i=1}^{r}f(x_{i}, k_{i}, d), \] where \(f(x,k,d)=x(x+d)\cdots(x+(k-1)d)\). In other words, they are interested whether the products of certain blocks of integers forming an arithmetic progression can be a square. There are many results in the paper. For example, the authors prove that for \(r = 4, d=1\) and \(k_i=5\) for \(i=1, 2, 3, 4\), the equation under consideration has infinitely many positive integer solutions. Other results are obtained for various combinations of integers \(d, r\) and \(k_i\). The proofs are elementary and are based on finding suitable substitution which reduces the problem to a simpler equation (mainly of Pell type) or just by finding polynomial parametric solution.
At the end of the paper many open question are formulated.
For Part II see [the author, Acta Math. Hung. 156, No. 1, 240–254 (2018; Zbl 1424.11078)].

MSC:

11D25 Cubic and quartic Diophantine equations
11D72 Diophantine equations in many variables
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References:

[1] Bauer, M.; Bennett, MA, On a question of Erdős and Graham, Enseign. Math., 53, 259-264 (2007) · Zbl 1155.11019
[2] Bennett, MA; Bruin, N.; Győry, K.; Hajdu, L., Powers from products of consecutive terms in arithmetic progression, Proc. Lond. Math. Soc., 92, 273-306 (2006) · Zbl 1178.11033 · doi:10.1112/S0024611505015625
[3] 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. · Zbl 1270.11026
[4] 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). · Zbl 0434.10001
[5] Erdős, P.; Selfridge, JL, The product of consecutive integers is never a power, Illinois J. Math., 19, 292-301 (1975) · Zbl 0295.10017 · doi:10.1215/ijm/1256050816
[6] Guy, R.K.: Unsolved Problems in Number Theory, 3rd edn. Springer-Verlag (Berlin 2004). · Zbl 1058.11001
[7] 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) · Zbl 1115.11020 · doi:10.4153/CMB-2004-037-1
[8] L. Hajdu, Sz. Tengely and R. Tijdeman, Cubes in products of terms in arithmetic progression, Publ. Math. Debrecen, 74 (2009), 215-232. · Zbl 1197.11038
[9] Hirata-Kohno, N.; Laishram, S.; Shorey, TN; Tijdeman, R., An extension of a theorem of Euler, Acta Arith., 129, 71-102 (2007) · Zbl 1137.11022 · doi:10.4064/aa129-1-6
[10] Katayama, S., Products of arithmetic progressions which are squares, J. Math. Tokushima Univ., 49, 9-12 (2015) · Zbl 1395.11046
[11] Luca, F.; Walsh, PG, On a Diophantine equation related to a conjecture of Erdős and Graham, Glas. Mat. Ser., III, 42, 281-289 (2007) · Zbl 1132.11319 · doi:10.3336/gm.42.2.03
[12] Obláth, R., Über das Produkt fünf aufeinander folgender Zahlen in einer arithmetischen Reihe, Publ. Math. Debrecen, 1, 222-226 (1950) · Zbl 0038.17901
[13] O. Rigge, Über ein diophantisches Problem, in: 9th Congress Math. Scand. (Helsingfors, 1938), Mercator (1939), pp. 155-160. · Zbl 0021.01003
[14] Skałba, M., Products of disjoint blocks of consecutive integers which are powers, Colloq. Math., 98, 1-3 (2003) · Zbl 1059.11030 · doi:10.4064/cm98-1-1
[15] Sz. Tengely, On a problem of Erdős and Graham, Period. Math. Hungar., 72 (2016), 23-28. · Zbl 1374.11058
[16] Sz. Tengely and M. Ulas, On products of disjoint blocks of arithmetic progressions and related equations, J. Number Theory, 165 (2016), 67-83. · Zbl 1373.11030
[17] Ulas, M., On products of disjoint blocks of consecutive integers, Enseign. Math., 51, 331-334 (2005) · Zbl 1112.11015
[18] B. Yıldız and E. Gürel, On a problem of Erdős and Graham, Bull. Braz. Math. Soc., New Series, 51 (2020), 397-415. · Zbl 1434.11080
[19] Y. Zhang, On products of consecutive arithmetic progressions. II, Acta Math. Hungar., 156 (2018), 240-254. · Zbl 1424.11078
[20] Zhang, Y.; Cai, T., On products of consecutive arithmetic progressions, J. Number Theory, 147, 287-299 (2015) · Zbl 1394.11027 · doi:10.1016/j.jnt.2014.07.003
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