An elementary solution of Lucas' problem (Q1801575)

In 1875 \textit{E. Lucas} [Problem 1180, Nouv. Ann. Math., II. Sér. 14, 336 (1875)] conjectured that the only nontrivial solution to the equation \(x(x+1) (2x+1)= 6y^ 2\) is \((x,y)=(24,70)\). The author proves by elementary methods that this is so. It has also been proved by more complicated methods [\textit{G. N. Watson}, Mess. Math. 48 (1918-1919), 1--22 (1919; JFM 46.0213.01) and \textit{W. Ljunggren}, Nordisk Mat. Tidskr. 34, 65--72 (1952; Zbl 0047.04102)].