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For all positive non-square integer multiplier k, there is an infinity ofmultiples of triangular numbers which are also triangular numbers. With asimple change of variables, these triangular numbers can be found usingsolutions of Pell equations. With some conditions on parities of fundamentalsolutions of the simple and generalized Pell equations, only odd solutions ofthe generalized Pell equation are retained to provide many infinitely solutionsfound on branches corresponding to each of the generalized fundamentalsolutions. General algebraic expressions of fundamental solutions of the Pellequations are found for some values of the multiplier k in function of theclosest natural square. Further, among the expressions of Pell equationsolutions, a set of recurrent relations is identical to those found previouslywithout the Pell equation solving method. It is found also that two constantsof the problem of multiples of triangular numbers are directly related to thefundamental solutions of the simple Pell equation, which is an unexpectedresult as it means that simple Pell equation fundamental solutions in allgenerality are related to constants in recurrent relations of the problem offinding triangular numbers multiple of other triangular numbers.

Triangular Numbers Multiple of Triangular Numbers and Solutions of Pell Equations