On certain consequences of the \((abc)\) conjecture in arithmetic and in logic (Q1359149)

Assuming the correctness of the \(abc\)-conjecture of Oesterlé-Masser, the author shows how to deduce the following result: Let \(x,y,d,d'\) be positive integers with \((x,d) =(y,d') =1\), and \((x,d) \neq(y,d')\), and such that \(x+id\) and \(y+id'\) have the same prime factors for \(1\leq i\leq 5\). Then \((x,y,d,d')\) belongs to a finite set. This conditional result can be improved when \(d\) and \(d'\) are fixed integers. It follows that the Erdös-Woods question as to the existence of an integer \(k>2\) such that for given positive integers \(x\) and \(y\), if \(x+i\) and \(y+i\) have the same prime factors for \(1\leq i\leq 5\), then \(x=y\), can be answered in the affirmative under the assumption of the \(abc\)-conjecture.