On linear combinations of products of consecutive integers
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ON LINEAR COMBINATIONS OF PRODUCTS OF CONSECUTIVE INTEGERS ´ A. BAZSO Institute of Mathematics, MTA-DE Research Group “Equations, Functions and Curves”, Hungarian Academy of Sciences and University of Debrecen, P.O. Box 400, H-4002 Debrecen, Hungary e-mail: [email protected] (Received January 10, 2020; revised May 22, 2020; accepted May 24, 2020)
Abstract. We investigate Diophantine problems concerning linear combinations of polynomials of the shape a0 x + a1 x(x + 1) + a2 x(x + 1)(x + 2) + · · · + an x(x + 1) . . . (x + n) with n ∈ N ∪ {0}. We provide effective finiteness results for the power, shifted power, and quadratic polynomial values of these linear combinations, generalizing the analogous results of Hajdu, Laishram and Tengely [10], and of B´erczes, Hajdu, Luca and the author [2] given for the sums x + x(x + 1) + x(x + 1)(x + 2) + · · · + x(x + 1) . . . (x + n), i.e., for the case a0 = a1 = · · · = an = 1. Our work is closely connected also with some results of Tengely and Ulas [15] concerning the case when the coefficients a0 , a1 , . . . , an are zeroes and ones.
1. Introduction For n = 0, 1, 2, . . ., let Pn (x) = x(x + 1) · · · (x + n), and put fn (x) =
n
Pk (x) =
k=0
k n
(x + j).
k=0 j=0
The first few such polynomials are: f0 (x) = x, f1 (x) = x + x(x + 1) = x2 + 2x = x(x + 2), f2 (x) = x + x(x + 1) + x(x + 1)(x + 2) = x3 + 4x2 + 4x = x(x + 2)2, f3 (x) = x(x + 2)(x2 + 5x + 5), f4 (x) = x(x + 2)(x3 + 9x2 + 24x + 17). The research of the author was supported in part by the Hungarian Academy of Sciences, and by the NKFIH grants ANN130909 and K128088. Key words and phrases: sum of products, block of consecutive integers, polynomial value. Mathematics Subject Classification: 11D41. c 2020 0236-5294/$ 20.00 © 0 Akad´ emiai Kiad´ o, ´ Budapest 0236-5294/$20.00 Akade ´miai Kiado , Budapest, Hungary
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´ ´ A. A. BAZS BAZSO O
In general, fn (x) is a monic polynomial with positive integer coefficients and of degree n + 1. These polynomials have been considered first by Hajdu, Laishram and Tengely [10]. They proved effective finiteness results on the Diophantine equation fn (x) = y m ,
(1)
and they gave all solutions to (1) when 1 ≤ n ≤ 10 such that n �= 2 if m = 2. They also proved that for n ≥ 3, all the roots of fn (x) are real and simple. Observe that equation (1) is closely related to several classical questions, e.g., to the power values of products of consecutive integers (see the fundamental paper of Erd˝ os and Selfridge [8]); to the problem of Erd˝os and Graham [7] concerning the power values of products of blocks Pn (x) for certain values of n (see also [1,4,11,14,17]); or to the power values of products of consecutive terms in an arithmetic progression studied among many others by Obl´ ath [12], Bennett, Bruin, Gy˝ory and Hajdu [3] and Gy˝ory, Hajdu and Pint´er [9]. Recently, Bazs´ o, B´erczes, Hajdu and Luca [2] considered the polynomial values of the polynomials fn (x). For a polynomial g(x) ∈ Q[x] with deg g �= 1 and n ≥ 3, they showed that the equation fn (x) =
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