On the Fundamental Arithmetical Structure and Distribution of Lucky Numbers
Marthinus Michael Dreeckmeier
TL;DR
This work develops a dedicated arithmetic theory for lucky numbers, a sieve-generated analogue of primes, by deriving an exact formula for the nth lucky number $l_n$ and establishing a Fundamental Theorem of Arithmetic for lucky numbers via a nonstandard binary operation $*$. It introduces a rich algebraic structure, including lucky divisors and new arithmetic functions $\overset{*}{\omega}$ and $\overset{*}{\Omega}$, and extends it with a left-extension $\circ$ to form a circ-d(n) divisor theory, together with fractional orders. The authors prove a Bertrand-type postulate for lucky numbers and obtain a strong bound on the gaps between consecutive lucky numbers, namely $l_{n+1}-l_n=O(\sqrt{l_n\log l_n})$, which implies a near-Legendre-scale density result $l_n \le x$ has a nearby successor within $O(\sqrt{x\log x})$. Collectively, these results lay a foundation for sieve-based arithmetic and open broad avenues for future research, including generalized Legendre-type conjectures and the infinitude of lucky primes.
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 $n$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 $n$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 best known bound for primes, and therefore obtain the first result that is known for lucky numbers but still conjectured to be true for primes. Together, these results establish a foundation for the arithmetic theory of lucky numbers and contribute to a broader understanding of sieve-generated sequences.