On the maximal length of two sequences of consecutive integers with the same prime divisors
For each integer \(n\geq 2\) let Supp(n) denote the set of prime factors of n. Suppose that for \(1\leq i\leq k\) we have \[ (1)\quad Supp(x+i)=Supp(y+i) \] where x,y,k are positive integers. A problem of P. Erdős and A. Woods is the following: Does there exist an integer \(k\geq 2\) such that (1) implies \(x=y?\) The authors prove that (1) implies that \[ (2)\quad \log k\leq C, (\log x\quad \log \log x)^{1/2}\quad for\quad x\geq 3 \] \[ (3)\quad y-x>\exp (C_ 2 k (\log k)^ 2 (\log \log k)^{-1})\quad for\quad k\geq 3 \] and \[ (4)\quad y-x>(k\quad \log \log y)^ D\quad for\quad y\geq 27 \] where \(D=C_ 3 k (\log \log y)\) (log log log y)\({}^{-1}\). Here \(C_ 1,C_ 2,C_ 3\) are effectively computable absolute positive constants. A powerful ingredient of the proof is an inequality of A. Baker dealing with the linear forms in the logarithms of rational numbers.
scientific article; zbMATH DE number 4128921
scientific article