[Paper Review] On Unique Additive Representations of Positive Integers and Some Close Problems

Vladimir Shevelev|ArXiv.org|Nov 3, 2008

semigroups and automata theory6 references3 citations

TL;DR

This paper introduces explicit formulas and recursive relations for Moser sequences $\{m_r(n)\}$, which uniquely represent every non-negative integer as $n = m_r(k) + r m_r(l)$, and studies the related sequence $s_n^{(r)} = r m_r(n-1) + 1$, showing its unique additive representation property and connections to the Josephus-Gro"er problem. The key contribution is a closed-form decomposition and recursion for these sequences, with applications to additive number theory and combinatorics.

ABSTRACT

Let, for r>=2, (m_r(n)),n>=0, be Moser sequence such that every nonnegative integer is the unique sum of the form s_k+rs_l. In this article we give an explicit decomposition formulas of such form and an unexpectedly simple recursion relation for Moser's numbers. We also study interesting properties of the sequence (rm_r(n-1)+1),n>=1, and its connection with some important problems. In particular, in the case of r=2 this sequence is surprisingly connected with the numbers solving the combinatorial Josephus-Groer problem. We pose also some open questions.

Motivation & Objective

• To derive explicit decomposition formulas for Moser sequences $\{m_r(n)\}$ that uniquely represent integers as $n = m_r(k) + r m_r(l)$.
• To establish simple recursive relations for $m_r(n)$, generalizing the known Moser-de Bruijn sequence for $r=2$.
• To investigate the sequence $s_n^{(r)} = r m_r(n-1) + 1$, showing its unique additive representation property for integers $n \equiv 1 \pmod{r}$, $n \geq r+1$, as $n = s_k^{(r)} + r s_l^{(r)}$.
• To explore connections between $s_n^{(2)}$ and the Josephus-Gro"er problem, revealing combinatorial significance.
• To examine the arithmetic and distributional properties of related sequences, including primality and composite density, and pose open problems on additive representations.

Proposed method

• Use of generating functions and functional equations: $f(x)f(x^r) = \frac{1}{1-x}$ for $m_r(n)$, and $f(x)f(x^r) = \frac{x^{r+1}}{1-x^r}$ for $s_n^{(r)}$, to define the sequences uniquely in the space of analytic functions with non-negative Taylor coefficients.
• Derivation of explicit formulas: $m_r(n) = \sum_{i \geq 0} \nu_i r^{2i}$, where $\nu_i$ are base-$r$ digits of $n$, and $s_n^{(r)} = 1 + \sum_{i \geq 0} \nu_i r^{2i+1}$, where $\nu_i$ are digits of $n-1$ in base $r$.
• Establishment of unique solutions to Diophantine equations: $m_r(k) + r m_r(l) = n$ and $s_k^{(r)} + r s_l^{(r)} = N$, with $k,l$ reconstructed from base-$r$ digit patterns of $n$ and $N-1$, respectively.
• Construction of arithmetic progressions $3a + 2bn$ that admit unique representations as $s_k^{(2)}(a,b) + 2 s_l^{(2)}(a,b)$, using generating function identities.
• Definition of a one-to-one mapping between integers $N \equiv 1 \pmod{r}$, $r+1 \leq N \leq r^{2t+1}+1$, and lattice points $[1,r^t] \times [1,r^t]$ via the coordinates $k,l$ from the unique decomposition.
• Use of digit-based reconstruction and modular arithmetic (e.g., $4^k \equiv 1 \pmod{3}$) to prove infinitude of composite numbers in sequences $a^{(c)}(n) = 2m_2(n) + c$.

Experimental results

Research questions

• RQ1 What explicit formulas and recursive relations govern the Moser sequences $m_r(n)$ that ensure unique additive representations $n = m_r(k) + r m_r(l)$?
• RQ2 How is the sequence $s_n^{(r)} = r m_r(n-1) + 1$ related to the Josephus-Gro"er problem, particularly for $r=2$?
• RQ3 Does every odd $c$ yield infinitely many composite numbers in the sequence $a^{(c)}(n) = 2m_2(n) + c$, and what does this imply for prime-based additive representations?
• RQ4 What is the minimal sequence $\{t_n\}$ that allows unique binary additive representation of all even integers, and what is its density?
• RQ5 What permutation of lattice points $[1,r^t] \times [1,r^t]$ minimizes or maximizes the Euclidean traveling salesman path?

Key findings

• The sequence $m_r(n)$ has an explicit formula: $m_r(n) = \sum_{i \geq 0} \nu_i r^{2i}$, where $\nu_i$ are the base-$r$ digits of $n$, enabling unique decomposition $n = m_r(k) + r m_r(l)$ with $k,l$ derived from even and odd-position digits of $n$.
• The sequence $s_n^{(r)} = r m_r(n-1) + 1$ satisfies $s_n^{(r)} = 1 + \sum_{i \geq 0} \nu_i r^{2i+1}$, where $\nu_i$ are digits of $n-1$ in base $r$, and every $N \equiv 1 \pmod{r}$, $N \geq r+1$, has a unique representation $N = s_k^{(r)} + r s_l^{(r)}$.
• For $r=2$, the sequence $s_n^{(2)}$ is directly connected to the Josephus-Gro"er problem, with terms matching sequences A088442 and A090569 in the OEIS.
• The sequence $a^{(c)}(n) = 2m_2(n) + c$ contains infinitely many composite numbers for any odd $c$, proven via modular arithmetic on digit patterns in base 4.
• The sequence $\{t_n\}$, formed by merging $\{s_n(1,1)\}$ and $\{s_n(1,2)\}$ with double repetition of 1, allows every even $N$ to be expressed as $N = t_k + t_l$, and is conjectured to be minimal in the sense that no sparser sequence can represent all even numbers uniquely.
• The mapping from integers $N \equiv 1 \pmod{r}$, $r+1 \leq N \leq r^{2t+1}+1$, to lattice points $[1,r^t] \times [1,r^t]$ via decomposition coordinates is bijective, with $r^{2t}$ such integers.