Proof of Erdős Problem #369: Consecutive Smooth Integers Near N

This Zenodo record contains the manuscript and source files for “Proof of Erdos Problem 369: Consecutive Smooth Integers Near N”. The paper studies strings of consecutive smooth integers near N and proves that for every ε > 0 and every integer k >= 2, every sufficiently large N admits a block of k consecutive N^ε-smooth integers in [N/2, N]. The main tools are cyclotomic factorization, Euler-totient estimates, and a multiplicative covering argument based on powers of 2ᴸ and 3ᴸ.