三角数的倍数与Pell方程的解
Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations
摘要：
对于所有正的非平方整数乘数k，存在无穷多个三角数的倍数，它们也是三角数。通过简单的变量变量变换，这些三角形数可以使用Pell方程的解来找到。通过对简单Pell方程和广义Pell方程的基本解的奇偶性给出一定的条件，仅保留广义Pell方程的奇解，从而在每个广义基本解对应的分支上找到许多无穷多解.对于乘子k为最近自然平方的函数的某些值，找到了Pell方程的基本解的一般代数式。此外，在Pell方程解的表达式中，一组递归关系与以前没有Pell方程求解方法时发现的那些递归关系相同。还发现三角数的倍数问题的两个常数与简单Pell方程的基本解直接相关，这是一个意想不到的结果，因为这意味着简单Pell方程的基本解在所有的一般性中与三角数的倍数问题的递归关系中的常数相关。
英文摘要：
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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