Numbers with a Large Prime Factor IV

This is an ‘approximation’ to the original question. Here of course φ is to be made as large as possible. Increasing values of φ for which such a p can be shown to exist have been provided by Ramachandra [13, 14], Graham [4], Baker [1], Jia [7, 8, 9, 10], Liu [11], Baker and Harman [2], Liu and Wu [17] and Harman [6]. In Harman’s book, the value of φ is 0.74, and it is noted that recent work on exponential sums due to Wu [17] and Robert and Sargos [16] give room for further progress. In the present paper we pursue this programme, and prove the following result. We write P (n) for the largest prime factor of a natural number n, and Q(n) for the smallest prime factor of n, with Q(1) = 1.