On the Fundamental Arithmetical Structure and Distribution of Lucky Numbers
Marthinus Michael Dreeckmeier

TL;DR
This paper develops an elementary number theory framework for lucky numbers, deriving an exact formula for their nth term, establishing an analogue of the Fundamental Theorem of Arithmetic, and analyzing their distribution and gaps.
Contribution
It introduces a new arithmetical structure for lucky numbers, including an exact formula, fundamental theorem analogue, and bounds on gaps, advancing their theoretical understanding.
Findings
Derived an exact formula for the nth lucky number.
Proved an analogue of the Fundamental Theorem of Arithmetic for lucky numbers.
Established a stronger asymptotic bound on gaps between lucky numbers.
Abstract
In this article, we will use elementary number theory techniques to investigate a sequence of integers defined by a sifting process called the lucky numbers. Ulam introduced lucky numbers as a sieve-based analogue of prime numbers. We derive an exact formula for the th lucky number, providing a new tool for quantitative analysis. We formulate and prove a version of the Fundamental Theorem of Arithmetic for lucky numbers. This theorem provides an entirely new viewpoint on number theory. Building on the fundamental theorem, we introduce foundational definitions and analogues of arithmetical functions. Additionally, we prove an analogue of Bertrand's postulate for lucky numbers. Finally, we use the formula for the th lucky number to prove a new result on the order of magnitude of the gaps between consecutive lucky numbers. We obtain an asymptotic bound that is much stronger than the…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAnalytic Number Theory Research · Advanced Mathematical Theories and Applications · Advanced Mathematical Theories
