On blocks of arithmetic progressions with equal products
Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 1, p. 183-199

Let f(X)[X] be a monic polynomial which is a power of a polynomial g(X)[X] of degree μ2 and having simple real roots. For given positive integers d 1 ,d 2 ,,m with <m and gcd(,m)=1 with μm+1 whenever m<2, we show that the equation f(x)f(x+d 1 )f(x+(k-1)d 1 )=f(y)f(y+d 2 )f(y+(mk-1)d 2 ) with f(x+jd 1 )0 for 0j<k has only finitely many solutions in integers x,y and k1 except in the case m=μ=2,=k=d 2 =1,f(X)=g(X),x=f(y)+y.

Soit f(X)[X] un polynôme qui est une puissance d’un polynôme g(X)[X] de degré μ2 et dont les racines réelles sont simples. Etant donnés les entiers positifs d 1 ,d 2 ,,m satisfaisant <m pgcd(,m)=1 et μm+1 si m<2, nous démontrons que l’équation f(x)f(x+d 1 )f(x+(k-1)d 1 )=f(y)f(y+d 2 )f(y+(mk-1)d 2 ) avec f(x+jd 1 )0 pour 0j<k ne possède qu’un nombre fini de solutions en les entiers x,y et k1, excepté dans le cas m=μ=2,=k=d 2 =1,f(X)=g(X),x=f(y)+y.

@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},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {1},
     year = {1997},
     pages = {183-199},
     zbl = {0889.11010},
     mrnumber = {1469667},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_1997__9_1_183_0}
}
Saradha, N. On blocks of arithmetic progressions with equal products. Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 1, pp. 183-199. http://www.numdam.org/item/JTNB_1997__9_1_183_0/

[1] A. Baker, Bounds for the solutions of the hyperelliptic equation, Proc. Camb. Phil. Soc. 65 (1969), 439-444. | MR 234912 | Zbl 0174.33803

[2] A. Brauer and G. Ehrlich, On the irreducibility of certain polynomials, Bull.Amer. Math. Soc. 52 (1946), 844-856. | MR 17750 | Zbl 0060.04705

[3] H.L. Dorwart and O. Ore, Criteria for the irreducibility of polynomials, Ann. of Math. 34 (1993), 81-94. | JFM 59.0906.03 | MR 1503098 | Zbl 0006.00406

[4] N. Saradha, T.N. Shorey and R. Tijdeman, On arithmetic progressions with equal products, Acta Arithmetica 68 (1994), 89-100. | MR 1302510 | Zbl 0812.11023

[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 1346806 | Zbl 0837.11015

[6] T.N. Shorey, London Math. Soc. Lecture Note Series, Number Theory, Paris 1992-3, éd. Sinnou David, 215 (1995), 231-244. | MR 1345182 | Zbl 0829.11015