The equation \(xyz=x+y+z=1\) in integers of a cubic field (Q1262332)

Cassels and Sierpiński showed that there are no rational solutions of the diophantine equation \[ (1)\quad xyz=x+y+z=1. \] Small studied the equation over finite fields, and Mollin et al. investigated the equation over quadratic number fields, finding finitely many such fields K (in fact, \({\mathbb{Q}}(i)\), \({\mathbb{Q}}(\sqrt{2})\), \({\mathbb{Q}}(\sqrt{5}))\) in which there does exist a solution for integer units \(u_ 1,u_ 2,u_ 3\) of K, of the equation \(u_ 1u_ 2u_ 3=u_ 1+u_ 2+u_ 3.\) In this note, we resolve completely the question of finding all cubic number fields whose ring of integers contains a solution to (1). The result is as follows. Theorem: Let K be a cubic number field with ring of integers \({\mathfrak O}_ K\). Then the equation (1) is solvable for x,y,z\(\in {\mathfrak O}_ K\) in precisely the two following instances. (i) \(K={\mathbb{Q}}(\theta)\), \(\theta^ 3-\theta^ 2-4\theta -1=0\), with solution (up to permutation) \((x,y,z)=(\theta,\theta^ 2-2\theta -2,- \theta^ 2+\theta +3)\); (ii) \(K={\mathbb{Q}}(\phi)\), \(\phi^ 3-\phi -1=0\), with solution (up to permutation) \((x,y,z)=(\phi +1,-\phi^ 2+1,\phi^ 2-\phi -1)\).