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Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations
Number Theory · Submitted 25 Feb 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim For all positive non-square integer multiplier k, there is an infinity of multiples of triangular numbers which are also triangular numbers.
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The abstract, sentence by sentence
For all positive non-square integer multiplier k, there is an infinity of multiples of triangular numbers which are also triangular numbers. With a simple change of variables, these triangular numbers can be found using solutions of Pell equations. With some conditions on parities of fundamental solutions of the simple and generalized Pell equations, only odd solutions of the generalized Pell equation are retained to provide many infinitely solutions found on branches corresponding to each of the generalized fundamental solutions. General algebraic expressions of fundamental solutions of the Pell equations are found for some values of the multiplier k in function of the closest natural square. Further, among the expressions of Pell equation solutions, a set of recurrent relations is identical to those found previously without the Pell equation solving method. It is found also that two constants of the problem of multiples of triangular numbers are directly related to the fundamental solutions of the simple Pell equation, which is an unexpected result as it means that simple Pell equation fundamental solutions in all generality are related to constants in recurrent relations of the problem of finding triangular numbers multiple of other triangular numbers.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For all positive non-square integer multiplier k, there is an infinity of multiples of triangular numbers which are also triangular numbers. | result 0.98 | 0.62 | 0.78 | asserted 1.00 |
| s1 | With a simple change of variables, these triangular numbers can be found using solutions of Pell equations. | method 0.96 | 0.67 | 0.08 | asserted 0.97 |
| s2 | With some conditions on parities of fundamental solutions of the simple and generalized Pell equations, only odd solutions of the generalized Pell equation are retained to provide many infinitely solutions found on branches corresponding to each of the generalized fundamental solutions. | method 0.73 | 0.85 | 0.08 | asserted 0.99 |
| s3 | General algebraic expressions of fundamental solutions of the Pell equations are found for some values of the multiplier k in function of the closest natural square. | result 0.83 | 0.91 | 0.07 | asserted 1.00 |
| s4 | Further, among the expressions of Pell equation solutions, a set of recurrent relations is identical to those found previously without the Pell equation solving method. | result 0.96 | 0.82 | 0.08 | asserted 1.00 |
| s5 | It is found also that two constants of the problem of multiples of triangular numbers are directly related to the fundamental solutions of the simple Pell equation, which is an unexpected result as it means that simple Pell equation fundamental solutions in all generality are related to constants in recurrent relations of the problem of finding triangular numbers multiple of other triangular numbers. | result 0.52 | 0.93 | 0.44 | asserted 1.00 |
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Original abstract
For all positive non-square integer multiplier k, there is an infinity of multiples of triangular numbers which are also triangular numbers. With a simple change of variables, these triangular numbers can be found using solutions of Pell equations. With some conditions on parities of fundamental solutions of the simple and generalized Pell equations, only odd solutions of the generalized Pell equation are retained to provide many infinitely solutions found on branches corresponding to each of the generalized fundamental solutions. General algebraic expressions of fundamental solutions of the Pell equations are found for some values of the multiplier k in function of the closest natural square. Further, among the expressions of Pell equation solutions, a set of recurrent relations is identical to those found previously without the Pell equation solving method. It is found also that two constants of the problem of multiples of triangular numbers are directly related to the fundamental solutions of the simple Pell equation, which is an unexpected result as it means that simple Pell equation fundamental solutions in all generality are related to constants in recurrent relations of the problem of finding triangular numbers multiple of other triangular numbers.
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States the same result 1 paper
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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.
Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers · 0.87
What it cites · 4 sentences without a confident label
Compares with 2 papers
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or by Chebyshev polynomials of the first kind $T_n-1(x_f)$ and of the second kind $U_n-2(x_f)$, evaluated at $x_f$ (see [ref] ), [equation] One notices that the second recurrent relations ([ref]) is similar in form to recurrent relations ([ref]) and ([ref]) found in [ref] .
Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials · Pell equations: A Reminder · 0.93
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or by Chebyshev polynomials of the first kind $T_n-1(x_f)$ and of the second kind $U_n-2(x_f)$, evaluated at $x_f$ (see [ref] ), [equation] One notices that the second recurrent relations ([ref]) is similar in form to recurrent relations ([ref]) and ([ref]) found in [ref] .
Recurrent Relations for Multiple of Triangular Numbers being Triangular Numbers · Pell equations: A Reminder · 0.93
Cites as background 1 paper
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In this paper, we investigate triangular numbers $T_ξ$ that are multiples of other triangular numbers $T_t$ [equation] Several authors have investigated this Diophantine equation; see, e.g., [ref] .
When are Multiples of Polygonal Numbers again Polygonal Numbers? · Introduction · 0.92
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