On Zeckendorf-Niven numbers and arithmetic progressions

Technical Summary: On Zeckendorf-Niven Numbers and Arithmetic Progressions

Problem Statement
This paper investigates the distribution of Zeckendorf-Niven and Lucas-Niven numbers within arithmetic progressions. A positive integer $n$ is defined as Zeckendorf-Niven if it is divisible by $s_Z(n)$, the number of nonzero summands in its unique Zeckendorf decomposition (a representation of $n$ as a sum of non-consecutive Fibonacci numbers $F_i$ with $i \ge 2$). Similarly, $n$ is Lucas-Niven if it is divisible by $s_L(n)$, the number of summands in its unique Lucas decomposition (using Lucas numbers $L_i$ with $i \ge 0$).

While the distribution of standard base-$b$ Niven numbers in arithmetic progressions is well-studied (e.g., the existence of infinitely many such numbers and bounds on consecutive terms), the behavior of these Fibonacci-based variants in arithmetic progressions had not been fully characterized. Specifically, the paper addresses whether every arithmetic progression contains infinitely many such numbers and establishes bounds on the maximum length of consecutive terms sharing these properties.

Methodology
The authors employ number-theoretic properties of Fibonacci and Lucas sequences, particularly their periodicity modulo $d$ (Pisano periods) and divisibility rules.

1. Periodicity and Divisibility: The proofs rely heavily on the fact that the Fibonacci sequence $(F_i)$ and Lucas sequence $(L_i)$ are periodic modulo any integer $d$. The authors utilize specific properties of Pisano periods, such as $\pi(F_{2j}) = 4j$ and $\pi(F_{2j+1}) = 8j+4$, to construct terms with specific properties.
2. Construction of Terms: To prove the existence of infinitely many Niven numbers in an arithmetic progression $(a + kd)$, the authors construct specific terms $N$ within the progression. They manipulate the Zeckendorf or Lucas decompositions by adding large Fibonacci/Lucas numbers that are multiples of the common difference (or related periods). This allows them to control the number of summands $s_Z(N)$ or $s_L(N)$ while ensuring the resulting number remains in the progression.
3. Modular Arithmetic and Case Analysis: For the Lucas-Niven case, the lack of direct divisibility properties (like $F_i | F_j$ if $i|j$) in the Lucas sequence requires a more nuanced approach. The authors use Dirichlet's Theorem on primes to select a prime $p \equiv -1 \pmod{10\pi_L(d)}$ and analyze cases based on the parity of indices in the decomposition to ensure the constructed number is divisible by $p$.
4. Local Analysis for Consecutive Terms: To bound the number of consecutive terms, the authors define truncated sums $z_6(n)$ (sums of Fibonacci components up to $F_6$) and analyze how the Zeckendorf decomposition changes when adding the common difference (e.g., $n \to n+2$). By tracking changes in the number of summands $s_Z(n)$, they derive contradictions for sequences longer than specific bounds.

Key Contributions and Results

• Infinitude in Arithmetic Progressions:

  • Theorem 2.6: Every arithmetic progression contains infinitely many Zeckendorf-Niven numbers. The proof constructs a term $N'$ in the progression such that $s_Z(N')$ divides $N'$.
  • Theorem 3.4: Every arithmetic progression contains infinitely many Lucas-Niven numbers. This result is achieved by adapting the construction technique to handle the specific modular constraints of the Lucas sequence.

• Distribution of Non-Niven Numbers:

  • Theorem 2.7: Every arithmetic progression contains arbitrarily long subsequences of consecutive non-Zeckendorf-Niven terms. This demonstrates that while Niven numbers are dense, they are not uniformly distributed.

• Bounds on Consecutive Terms in 2-APs:

  • Theorem 4.3: The maximum number of consecutive Zeckendorf-Niven terms in a 2-AP (common difference 2) is 8. The only such sequence of length 8 or more is the subsequence $2, 4, 6, 8, 10, 12, 14, 16, 18$.
  • Theorem 4.4: There exist sequences of five consecutive Zeckendorf-Niven terms in a 2-AP (e.g., $n, n+2, n+4, n+6, n+8$ where $n = 27 + F_{120j+17}$).

• Consecutive Terms with Constant Summand Count:

  • Theorem 5.1: In an arithmetic progression with a Fibonacci common difference $F_d$, there exist infinitely many starting points $a$ such that $s_Z(a) = s_Z(a+F_d) = s_Z(a+2F_d)$. However, it is impossible to have four consecutive terms with the same $s_Z$-value.
  • Theorem 5.2: For $d \ge 3$, there exist infinitely many $a$ such that $a, a+F_d, a+2F_d$ are all Zeckendorf-Niven and share the same $s_Z$-value.

Significance and Claims
The paper claims to extend the theory of Niven numbers from standard base representations to Zeckendorf and Lucas representations. The primary significance lies in establishing that, similar to standard Niven numbers, Zeckendorf-Niven and Lucas-Niven numbers are not rare; they appear infinitely often in every arithmetic progression.

Furthermore, the work provides precise structural limitations on the clustering of these numbers. While standard Niven numbers can have long runs of consecutive terms (up to 20 in 1-APs), the Zeckendorf-Niven numbers in 2-APs are strictly bounded (maximum 8). The paper also highlights a unique phenomenon in Fibonacci-difference progressions where the number of summands can remain constant for exactly three consecutive terms, a property that cannot be extended to four.

The authors note that these results rely on the unique decomposition properties of Fibonacci and Lucas numbers and the periodicity of these sequences modulo integers. The work does not propose experimental applications but rather contributes to the theoretical understanding of the distribution of integers defined by additive number-theoretic properties.