[Paper Review] Repdigits as products of consecutive balancing or Lucas-balancing numbers

S. G. Rayaguru, G. K. Panda|arXiv (Cornell University)|Dec 28, 2018

Advanced Mathematical Theories and Applications7 references4 citations

TL;DR

This paper investigates whether repdigits—numbers with repeated identical digits—can appear as balancing or Lucas-balancing numbers or as products of consecutive such numbers. Using modular arithmetic and properties of divisibility sequences, the authors prove that no balancing number (except B₁=1 and B₂=6) is a repdigit with digits other than 6, and no product of consecutive balancing or Lucas-balancing numbers forms a multi-digit repdigit. The only Lucas-balancing repdigits are C₁=3 and C₃=99.

ABSTRACT

Repdigits are natural numbers formed by the repetition of a single digit. In this paper, we explore the presence of repdigits in the product of consecutive balancing or Lucas-balancing numbers.

Motivation & Objective

• To determine whether repdigits (numbers with repeated digits) appear in the balancing or Lucas-balancing number sequences.
• To investigate whether products of consecutive balancing or Lucas-balancing numbers can form repdigits.
• To resolve open questions about the existence of repdigits in these sequences, particularly those with all digits equal to 6.
• To apply modular arithmetic and divisibility properties to rule out the existence of such repdigits beyond known small cases.

Proposed method

• Employing modular arithmetic by analyzing the balancing and Lucas-balancing sequences modulo small integers (3, 4, 5, 7, 8, 9, 11, 20) to derive periodic residue patterns.
• Using the strong divisibility property of the balancing sequence (Bₘ ∣ Bₙ iff m ∣ n) to derive contradictions when assuming a repdigit form.
• Reducing the Diophantine equations Bₙ = a·(10ᵐ−1)/9 and Cₙ = a·(10ᵐ−1)/9 modulo various integers to eliminate possible digit values a.
• Analyzing the product of consecutive terms BₙBₙ₊₁ and CₙCₙ₊₁⋯Cₙ₊ₖ modulo 5, 7, and 8 to show they cannot match the residue of any repdigit.
• Applying case analysis for each digit a ∈ {1, 2, ..., 9} to systematically rule out repdigit forms in both sequences.
• Using the fact that Lucas-balancing numbers are always odd to restrict possible values of a to {1, 3, 5, 7, 9} and further eliminate cases via modular constraints.

Experimental results

Research questions

• RQ1 Are there any repdigits (other than 1 and 6) in the balancing number sequence?
• RQ2 Can the product of two or more consecutive balancing numbers form a multi-digit repdigit?
• RQ3 Are there any Lucas-balancing numbers that are repdigits beyond C₁ = 3 and C₃ = 99?
• RQ4 Can the product of consecutive Lucas-balancing numbers form a repdigit with more than one digit?
• RQ5 Is it possible for a balancing number to be 6 times a repunit (e.g., 66, 666, etc.)?

Key findings

• The only balancing numbers that are repdigits are B₁ = 1 and B₂ = 6; no other balancing number is a repdigit with digits other than 6.
• The product of any two consecutive balancing numbers is never a multi-digit repdigit, although B₁B₂ = 6 is a single-digit repdigit.
• The only Lucas-balancing numbers that are repdigits are C₁ = 3 and C₃ = 99; no larger Lucas-balancing number is a repdigit.
• The product of any sequence of consecutive Lucas-balancing numbers is never a repdigit with more than one digit.
• For a balancing number to be a repdigit with all digits 6, it must satisfy n ≡ 14 (mod 96) and m ≡ 1 (mod 6), but no such number is known beyond B₂ = 6.
• The paper leaves open the question of whether any balancing number is 6 times a repunit (e.g., 66, 666), though no such number is found in the first 200 terms.