On products of consecutive arithmetic progressions. III
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ON PRODUCTS OF CONSECUTIVE ARITHMETIC PROGRESSIONS. III Y. ZHANG School of Mathematics and Statistics, Changsha University of Science and Technology; Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha 410114, People’s Republic of China e-mail: [email protected] (Received March 12, 2020; revised August 13, 2020; accepted August 22, 2020)
Abstract. Let f (x, k, d) = x(x + d) · · · (x + (k − 1)d) be a polynomial with k ≥ 2, d ≥ 1. We consider the Diophantine equation ri=1 f (xi , ki , d) = y 2 , r ≥ 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].
1. Introduction Let us define the polynomial f (x, k, d) = x(x + d) · · · (x + (k − 1)d) with k ≥ 2, d ≥ 1. Many authors have studied the Diophantine equation (1.1)
r
f (xi , ki , d) = y 2 ,
i=1
where r ≥ 1, f (xi , ki , d) = xi (xi + d) · · · (xi + (ki − 1)d) are disjoint for i = 1, . . . , r, and 2 ≤ k1 ≤ k2 ≤ · · · ≤ kr . (1) The case r = 1, d ≥ 1. There are many results about (1.1) and the more general Diophantine equation f (x, k, d) = by l , 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). Key words and phrases: Diophantine equation, consecutive arithmetic progression, positive integer solution, Pell equation. Mathematics Subject Classification: primary 11D25, secondary 11D72.
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Y. ZHANG
where b > 0, l ≥ 3 and the greatest prime factor of b does not exceed k; we can refer to [2,5–9,12,13]. (2) The case r ≥ 2, d = 1. When r = 2, d = 1, ki = 3, Sastry [6] showed that (1.1) has infinitely many positive integer solutions (x1 , x2 , y), where x1 , x2 satisfying x2 = 2x1 − 1 and (x1 + 1)(2x1 − 1) is a square. Erd˝ os and Graham [4, p. 67] asked whether (1.1) has, for fixed r ≥ 1, d = 1 and k1 , k2 , . . ., kr with ki ≥ 4 for i = 1, 2, . . . , r, at most finitely many positive integer solutions (x1 , x2 , . . . , xr , y) with xi + ki − 1 < xi+1 for 1 ≤ i ≤ r − 1. Skalba [14] obtained a bound for the smallest solution and estimated the number of solutions below a given bound. Ulas [17] answered the above question of Erd˝ os and Graham in the negative when either r = 4, d = 1, ki = 4, i = 1, 2, 3, 4, or r ≥ 6, d = 1, ki = 4, 1 ≤ i ≤ r. Bauer and Bennett [1] extended Ulas’s result to the cases r = 3 and r = 5. For the case r = 2, d = 1, k1 = k2 = 4, (1.1) has a positive integer solution (x1 , x2 , y) = (33, 1680, 3361826160). Luca and Walsh [11] studied this case by using the identity (x − 1)x(x + 1)(x + 2) = (x2 + x − 1)2 −
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