On blocks of arithmetic progressions with equal products

Saradha, N.

Saradha, N.

Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 183-199.

Résumé
Abstract

Soit $f (X) \in ℚ [X]$ un polynôme qui est une puissance d’un polynôme $g (X) \in ℚ [X]$ de degré $μ \geq 2$ et dont les racines réelles sont simples. Etant donnés les entiers positifs $d_{1}, d_{2}, ℓ, m$ satisfaisant $ℓ < m$ pgcd $(ℓ, m) = 1$ et $μ \leq m + 1$ si $m < 2$ , nous démontrons que l’équation

f (x) f (x + d_{1}) \dots f (x + (ℓ k - 1) d_{1}) = f (y) f (y + d_{2}) \dots f (y + (m k - 1) d_{2})

avec

f (x + j d_{1}) \neq 0

pour

0 \leq j < ℓ k

ne possède qu’un nombre fini de solutions en les entiers

x, y

k \geq 1

, excepté dans le cas

m = μ = 2, ℓ = k = d_{2} = 1, f (X) = g (X), x = f (y) + y .

Let $f (X) \in ℚ [X]$ be a monic polynomial which is a power of a polynomial $g (X) \in ℚ [X]$ of degree $μ \geq 2$ and having simple real roots. For given positive integers $d_{1}, d_{2}, ℓ, m$ with $ℓ < m$ and gcd $(ℓ, m) = 1$ with $μ \leq m + 1$ whenever $m < 2$ , we show that the equation

f (x) f (x + d_{1}) \dots f (x + (ℓ k - 1) d_{1}) = f (y) f (y + d_{2}) \dots f (y + (m k - 1) d_{2})

with

f (x + j d_{1}) \neq 0

for

0 \leq j < ℓ k

has only finitely many solutions in integers

x, y

and

k \geq 1

except in the case

m = μ = 2, ℓ = k = d_{2} = 1, f (X) = g (X), x = f (y) + y .

MR Zbl

@article{JTNB_1997__9_1_183_0,
     author = {Saradha, N.},
     title = {On blocks of arithmetic progressions with equal products},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {183--199},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {1},
     year = {1997},
     mrnumber = {1469667},
     zbl = {0889.11010},
     language = {en},
     url = {https://numdam.org/item/JTNB_1997__9_1_183_0/}
}

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Saradha, N. On blocks of arithmetic progressions with equal products. Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 1, pp. 183-199. https://numdam.org/item/JTNB_1997__9_1_183_0/

Bibliographie
Cité par

[1] A. Baker, Bounds for the solutions of the hyperelliptic equation, Proc. Camb. Phil. Soc. 65 (1969), 439-444. | MR | Zbl

[2] A. Brauer and G. Ehrlich, On the irreducibility of certain polynomials, Bull.Amer. Math. Soc. 52 (1946), 844-856. | MR | Zbl

[3] H.L. Dorwart and O. Ore, Criteria for the irreducibility of polynomials, Ann. of Math. 34 (1993), 81-94. | JFM | MR | Zbl

[4] N. Saradha, T.N. Shorey and R. Tijdeman, On arithmetic progressions with equal products, Acta Arithmetica 68 (1994), 89-100. | MR | Zbl

[5] N. Saradha, T.N. Shorey and R. Tijdeman, On values of a polynomial at arithmetic progressions with equal products, Acta Arithmetica 72 (1995), 67-76. | MR | Zbl

[6] T.N. Shorey, London Math. Soc. Lecture Note Series, Number Theory, Paris 1992-3, éd. Sinnou David, 215 (1995), 231-244. | MR | Zbl