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

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     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 = {http://archive.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, Volume 9 (1997) no. 1, pp. 183-199. http://archive.numdam.org/item/JTNB_1997__9_1_183_0/

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