[Paper Review] Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers

Vladimir Pletser|arXiv (Cornell University)|Dec 31, 2020

Algebraic and Geometric Analysis3 references4 citations

TL;DR

This paper derives recurrent relations for triangular numbers that are multiples of other triangular numbers, showing that for any non-square positive integer multiplier k, infinitely many solutions exist. It establishes recurrence relations based on Pell equations and provides closed-form solutions, while for squared k, solutions are finite and depend on specific values of k.

ABSTRACT

We search for triangular numbers that are multiples of other triangular numbers. It is found that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers that are triangular numbers and recurrent relations are deduced theoretically. If the multiplier is a squared integer, there is either one or no solution, depending on the multiplier value.

Motivation & Objective

• To identify all triangular numbers T_ξ that are integer multiples kT_t of other triangular numbers T_t.
• To derive general recurrent relations for sequences of solutions (t_n, ξ_n) when k is a non-square positive integer.
• To analyze the structure of solution sequences by introducing the concept of 'rank' r, defined by the regularity of ratios t_n/t_{n−r}.
• To determine conditions under which solutions exist when k is a perfect square, showing only one or no solution exists depending on k.
• To provide closed-form expressions and recurrence relations for t_n, ξ_n, T_t, and T_ξ using linear recurrence with constant coefficients.

Proposed method

• Transform the Diophantine equation T_ξ = kT_t into the generalized Pell equation x² − ky² = 1 − k, where x = 2ξ + 1 and y = 2t + 1.
• Use the theory of Pell equations to prove the existence of infinitely many solutions when k is a non-square positive integer.
• Define the 'rank' r as the smallest integer such that the ratio t_n / t_{n−r} is approximately constant or decreases regularly, enabling recurrence of order 2r.
• Derive recurrence relations of the form X_n = 2(κ + 1)X_{n−r} − X_{n−2r} + κ for t_n and ξ_n, where κ = t_r + t_{r−1} = ξ_r − ξ_{r−1} − 1.
• Establish recurrence relations for λ_n and ξ_n using auxiliary sequences with initial conditions λ_{−1} = −1, λ_0 = 0, λ_1 = 1 and ξ_0 = 0, ξ_1 = t.
• Prove that for k = λ_n², the values k_n = λ_n² yield valid solutions to T_ξ = kT_t, and tabulate solutions for increasing λ_n and k < 10⁴.

Experimental results

Research questions

• RQ1 For which non-square positive integers k do there exist infinitely many triangular numbers T_ξ that are multiples kT_t of other triangular numbers T_t?
• RQ2 What is the structure of the solution sequences (t_n, ξ_n) for such k, and can they be described by a recurrence relation?
• RQ3 How can the concept of 'rank' r be formally defined and used to derive recurrence relations of order 2r for the solution sequences?
• RQ4 What conditions determine the existence of solutions when k is a perfect square, and how many solutions exist in such cases?
• RQ5 Can closed-form recurrence relations be derived for t_n, ξ_n, T_t, and T_ξ using the parameters κ and r?

Key findings

• For any positive non-square integer k, there are infinitely many solutions (ξ, t) such that T_ξ = kT_t, as guaranteed by the solvability of the associated Pell equation.
• The solution sequences (t_n, ξ_n) satisfy a second-order linear recurrence of the form X_n = 2(κ + 1)X_{n−r} − X_{n−2r} + κ, where r is the rank and κ is a constant derived from initial solution pairs.
• For k = λ_n², the values of k_n = λ_n² yield valid solutions, and the first five values of k_n and ξ_n are tabulated for t = 1 to 6, showing k_n are squares and triangular numbers when t = 1.
• When k is a perfect square, solutions exist only for specific k values; for example, k = 1, 36, 100, 196, 324, ..., and only one or no solution exists per k, as confirmed in Table 7.
• The recurrence for λ_n is λ_n = (4t + 2)λ_{n−1} − λ_{n−2}, with λ_{−1} = −1, λ_0 = 0, λ_1 = 1, and ξ_n follows ξ_n = (4t + 2)ξ_{n−1} − ξ_{n−2} + 2t, with ξ_0 = 0, ξ_1 = t.
• For t = 1, the sequence of k_n = λ_n² corresponds to OEIS A001110 (numbers that are both square and triangular), and ξ_n corresponds to OEIS A001108.