Solution of a diophantine problem (Q796564)

The author proves that the only positive integers which are simultaneously triangular (i.e., of the form \(y(y+1)/2)\) and pyramidal (i.e., of the form \(x(x+1)(2x+1)/6)\) are \(1, 55, 91\) and \(208\,335\). The result is obtained by determining all integer points on the elliptic curve \(3y^ 2=x^ 3-x+3\). The right hand side is decomposed in a suitable cubic field, and considering the residues of \(x \bmod 3\) reduces the problem to solving eight equations, a typical example being \(4\,U^ 4- 36\,UV^ 3+9\,V^ 4=1\). Then these eight equations are solved in integers using the arithmetic of quartic fields. The proofs are complete and fairly detailed.