Three triangular numbers contained in geometric progression
The authors formulate the Conjecture. Let \(m>1\) be a positive integer. The system of Diophantine equations \[ \begin{cases} x^2-(m^2-1)y^2=1\\ z^2-(m^2-1)y^r=1 \end{cases} \] has no positive integer solution \((x,y,r)\) with \(y>1\) and \(r>2\). They prove the Theorem. Let \(T_n:=\frac{n(n+1)}{2},n\in{\mathbb N}\) be the triangular numbers. If the conjecture holds, then \(T_{n_1},T_{n_2},T_{n_3}\) are three distinct terms in a geometric progression if and only if \((x,y,z)=(2n_1+1,2n_3+1,2n_2+1)\) is a solution of the diophantine equation \[ (x^2-1)(y^2-1)=(z^2-1)^2,\quad 1<x<z<y. \]
scientific article; zbMATH DE number 5693243