Pairs of equiperimeter and equiareal triangles whose sides are perfect squares (Q2154789)

It is known by previous work of others, in particular the work [\textit{A. Choudry}, Hardy-Ramanujan J. 30, 19--30 (2007; Zbl 1157.11010)] by the first author, that there are an arbitrarily large number of scalene rational triangles (namely, triangles with rational sides and area) with the same area and perimeter. In the paper under review the authors show that there are infinitely many pairs of rational triangles with the same area and perimeter and which additionally have squared sides. To win they need to manipulate the two given conditions of equal perimeter and area on triangles of sides \((a^2,b^2,c^2)\) and \((d^2,e^2,f^2)\). They choose \((a,b,c,d,e,f)\) to be certain linear combination of parameters \(p,q,r\) with coefficients \(\pm 1\) and another parameter \(u\). With their choice of parameters one of the equations is already satisfied and the other one reduces to a polynomial in \(p,q,r\) of degree \(3\) to be a square. They perform a computation in the range \(|p|+|q|+|r|\le 700\) and obtain a couple of numerical examples which satisfy all the required properties. Using one of their examples they construct an elliptic curve over \({\mathbb Q}[m]\) given in a quartic model, which for an appropriate specialisation of the parameter \(m\) yields an elliptic curve over \(\mathbb Q\) with a rational point \(P\) of infinite order. Conversely, each rational point of infinite order on that curve yields a pair of triangles with the desired property so the authors conclude by using the pairs of triangles corresponding to the multiples of \(P\) in the Mordell-Weil group of their elliptic curve. The paper concludes with a few more numerical examples which can be used to yield the same result.