Irreducibility of polynomials and arithmetic progressions with equal products of terms
Frits Beukers, T. N. Shorey, Rob Tijdeman · 1999
In some fundamental papers Davenport, Lewis and Schinzel [DLS], Schinzel [Sch1, Sch3] and Fried [Fr1, Fr2, Fr3] have shown how irreducibility criteria for polynomials f(X) g(Y ) in combination with results of Runge or Siegel can be used to prove the niteness of the solutions of the corresponding diophantine equation f(x) = g(y) in integers x; y. In the present paper we are particularly interested in the case f(X) = X(X+d1 ) (X+(m 1)d1 ), g(Y ) = Y (Y +d2) (Y +(n 1)d2 ), i.e. the diophantine equation x(x + d1 ) (x + (m 1)d1) = y(y + d2 ) (y + (n 1)d2 ): (1) We rst give some history on this equation and indicate how results for this equation can be derived from general irreducibility theory in the literature. Then we give direct proofs of the results using only basic facts on algebraic curves. 1 When do nite arithmetic sequences have equal products of terms? The question, under the restriction that the arithmetic progressions have equal lengths, was posed in Poland by Gabovich in 1966 [Ga]. He mentioned the example 2 6 10 = 4 5 6 and gave an innite class of examples of length 4 including 7 20 33 46 = 20 21 22 23 and 18 37 56 75 = 24 37 50 63. Some innite classes of solutions of length 5 were given by Szymiczek [Sz] and Choudhry [Ch]. In 1968 Makowski [Ma] observed that for every positive integer m 2 6 10 (4m 2) = (m + 1)(m + 2) (2m): (1) An opposite result was obtained by Saradha, Shorey and Tijdeman [SaST1]. 1 2 Theorem A. For xed integers d 1 > d 2 > 0 there are only nitely many positive integers m > 2, x; y gcd(x; y; d 1 ; d 2 ) = 1 and x(x + d 1 ) (x + (m 1)d 1 ) = y(y + d 2 ) (y + (m 1)d 2 ) (2) except for the solutions (1). The other solutions are eectively comput...