Recent zbMATH articles in MSC 11Y05https://www.zbmath.org/atom/cc/11Y052021-04-16T16:22:00+00:00WerkzeugPerfect triangles on the curve \(C_4\).https://www.zbmath.org/1456.112362021-04-16T16:22:00+00:00"Ismail, Shahrina"https://www.zbmath.org/authors/?q=ai:ismail.shahrinaA triangle with the three sides \((a,b,c)\), the three medians \((k,l,m)\) and the area \(\Delta\) is called a Heron triangle if \(a,b,c\) and \(\Delta\) are rational. If each of the \(7\) mentioned parameters is rational, the triangle is called perfect. While finding a perfect triangle is still an open problem, it has been shown in previous work that Heron triangles with two rational medians can be parametrized by eight curves \(C_1,\ldots, C_8\). The main contribution of this paper is to prove that no perfect triangle arises from the curve \(C_4\), except possibly for one case that could not have been eliminated by the proposed methodology, and thus remains a prospect for future work.
The author starts by showing that finding a perfect triangle arising from \(C_4\) is equivalent to finding integers \(n\) such that \(Z(nP)\) is square, where \(Z(x,y)=R(x)-S(x)y\) for two explicitly defined polynomials \(R\) and \(S\), and \(P\) is an infinite-order generator of a certain elliptic curve. This consideration leads to a set \(Y\) of small candidates for \(n\), for which it is checked that no perfect triangle arises from them. Based on the elements of \(Y\), restrictions are imposed on the lifting to larger multiples of the point \(P\). This technique, which may be considered as an inductive method for the Mordell-Weil sieve, allows to eliminate all candidates for \(n\) except those relating to the value \(-2\in Y\). The author also mentions that a perfect triangle arising from this exceptional case would have to have side lengths at least \(10^{10^{10}}\).
The main result of the paper is an interesting and valuable contribution to the open problem of the existence of perfect triangles. With regard to the readability of the paper, I believe that it would have benefited from more detailed explanations of the ideas and the strategy. I also noticed some minor inconsistencies. For example, the claim about \(Z(nP)\) being square appears to be missing in the formulation of Theorem 2.1, and the formulation of Theorem 3.4 appears to contradict what it actually should say in order to conform with Table 1 and its application in the proof of Theorem 6.1. However, most parts of the paper are well-written, in particular the presentation of the problem and the state-of-the-art in the introduction.
Reviewer: Markus Hittmeir (Wien)