On the positive integral solutions of the Diophantine equation \(x^3+by+1-xyz=0\) (Q957531)

The authors prove that the title Diophantine equation has at most \(11b+o(b)\) solutions \((x,y,z)\) for fixed \(b\). This is a weak form of a conjecture due to \textit{S. P. Mohanty} and \textit{A. M. S. Ramasamy} [see Bull. Malays. Math. Soc. (2) 7, 23--28 (1984; Zbl 0547.10014)]. Solutions to the title equation fulfill a divisibility condition namely \(xz-b\mid \frac{b^3+z^3}{\gcd(b,z)^2}\) with at most three possible (explicit known) exceptions. By a careful estimation of the sum of divisors \(\sum \tau(z_1^3+b_1^3)\) with \(z=dz_1\), \(b=db_1\) and \(d=\gcd (b,z)\) the authors are able to prove their result. Note \(\tau(x)\) denotes the number of divisors of \(x\).