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This research addresses the existence of consecutive smooth integers within a specific range near large integers N.

April 3, 2026Open Access

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

Key Points

This research addresses the existence of consecutive smooth integers within a specific range near large integers N.
Proved the existence of k consecutive N^ε-smooth integers in the interval [N/2, N].
Utilized cyclotomic factorization to understand integer decompositions.
Applied Euler-totient estimates for analyzing smoothness.
Employed a multiplicative covering argument informed by the powers of 2 and 3.
Demonstrated that for every ε > 0 and k >= 2, sufficient N yields k consecutive smooth integers.
Confirmed results using theoretical frameworks involving factorization and estimates.

Abstract

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.

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

Key Points

Abstract

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