On power values of pyramidal numbers. I (Q2919682)

Let \(m\geq 3\) be a fixed integer and consider the pyramidal numbers \(\mathrm{Pyr}_m(x)=\frac 16 x(x+1)((m-2)x+5-m)\) of order \(m\). The paper under review treats the problem to find all square pyramidal numbers. Therefore the authors consider the family of elliptic curves NEWLINE\[NEWLINE E_m\; :\; y^2 =\frac 16 x(x+1)((m-2)x+5-m).NEWLINE\]NEWLINE Assume that \(k\) is a positive integer such that \(2k^4-1\) is square and \(k^4-1=2^st\), with \(t\) odd and square free. If \(m=3k^4+2\) and \(E_m(\mathbb Q)\) has rank \(1\), then all integer points on \(E_m\), with \(y\geq 0\) are \((0,0),(-1,0),(1,1)\) and \((2k^4-2,2k^8-3k^4+1)\).