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Three triangular numbers contained in geometric progression. (English) Zbl 1206.11037
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 $(x^2-1)(y^2-1)=(z^2-1)^2,\quad 1<x<z<y.$
##### MSC:
 11D25 Cubic and quartic Diophantine equations 11D41 Higher degree equations; Fermat’s equation
##### Keywords:
triangular numbers; geometric progressions
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