## On blocks of arithmetic progressions with equal products.(English)Zbl 0889.11010

Let $$g\in \mathbb{Q} [X]$$ be a monic polynomial of degree $$\mu\geq 2$$ with simple real roots, and let $$f(X)= g^b(X)$$ with $$b\in \mathbb{N}$$. For given positive integers $$d_1, d_2, \ell,m$$ with $$\ell <m$$ and coprime, and $$\mu\leq m+1$$ whenever $$m>2$$, it is shown that the diophantine equation $f(x) f(x+d_1) \dots f\bigl(x+ (\ell k-1) d_1\bigr) =f(y) f(y+d_2) \dots f\bigl(y+ (\ell k-1) d_2 \bigr),$ with $$f(x+j d_1)\neq 0$$ for $$0\leq j< \ell k$$, has only finitely many solutions in integers $$x,y$$ and $$k\geq 1$$, except in the case $$m= \mu =2$$, $$b= \ell= k=d_2 =1$$, $$x= f(y)+y$$. This effective (but not explicit) theorem extends a joint result of N. Saradha, T. N. Shorey and R. Tijdeman [Acta Arith. 72, No. 1, 67-76 (1995; Zbl 0837.11015)].

### MSC:

 11D61 Exponential Diophantine equations

Zbl 0837.11015
Full Text:

### References:

 [1] Baker, A., Bounds for the solutions of the hyperelliptic equation, Proc. Camb. Phil. Soc.65 (1969), 439-444. · Zbl 0174.33803 [2] Brauer, A. and Ehrlich, G., On the irreducibility of certain polynomials, Bull.Amer. Math. Soc.52 (1946), 844-856. · Zbl 0060.04705 [3] Dorwart, H.L. and Ore, O., Criteria for the irreducibility of polynomials, Ann. of Math.34 (1993), 81-94. · JFM 59.0906.03 [4] Saradha, N., Shorey, T.N. and Tijdeman, R., On arithmetic progressions with equal products, Acta Arithmetica68 (1994), 89-100. · Zbl 0812.11023 [5] Saradha, N., Shorey, T.N. and Tijdeman, R., On values of a polynomial at arithmetic progressions with equal products, Acta Arithmetica72 (1995), 67-76. · Zbl 0837.11015 [6] Shorey, T.N., London Math. Soc. Lecture Note Series, Number Theory, Paris1992-3, éd. Sinnou David, 215 (1995), 231-244. · Zbl 0829.11015
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