On the maximal length of two sequences of integers in arithmetic progressions with the same prime divisors
It was conjectured by Erdös and Woods [see \textit{R. Balasubramanian}, \textit{T. N. Shorey}, \textit{M. Waldschmidt}, Acta Math. Hung. 54, 225-236 (1989; Zbl 0689.10049)] that there exists a positive integer \(k\) with the property: if \(x,y\in \mathbb{Z}^+\) are such that for \(1\leq i\leq k\) the sets of prime divisors of \(x+i\) and \(y+i\) coincide, then \(x=y\). In the present paper an analogue of this problem for arithmetic progressions is considered. Let \(u(n)\) denote the greatest squarefree divisor of the positive integer \(n\). For each quadruple \((x,y,d,d')\) of positive integers satisfying \[ (x,d)\neq(y,d') \quad \text{and} \quad \text{gcd}(x,d) = \text{gcd}(y,d')=1, \tag{1} \] let \(K=K(x,y,d,d')\) be the largest positive integer \(K\) for which \(u(x+id) = u(y+id')\) for \(0\leq i\leq K-1\). The authors prove the following propositions, assuming the \(abc\)-conjecture of Masser and Oesterlé to be true: For each pair \((d,d')\) of two positive integers, the set of pairs \((x,y)\) satisfying (1) and \(K(x,y,d,d')\)\(>2\) is finite; and the set of quadruples \((x,y,d,d')\) satisfying (1) and \(K(x,y,d,d')>4\) is finite. Also some interesting partial, unconditional results are obtained.
scientific article; zbMATH DE number 889761