Triangular Numbers in Geometric Progression

In [R. K. Guy, Unsolved Problems in Number Theory, 3rd ed. Springer Verlag, New York, 2004, D23], it is stated that Sierpinski asked the question of whether or not there exist four (distinct) triangular numbers in geometric progression. Szymiczek conjectured that the answer is negative. Recently M. A. Bennett [Integers: Electronic Journal of Combinatorial Number Theory 5(1) (2005)] proved that there do not exist four distinct triangular numbers in geometric progression with the common ratio being a positive integer. In this paper we prove that there do not exist four distinct triangular numbers in geometric progression. Thus Sierpinski’s question is answered and Szymiczek’s conjecture is confirmed.