Let \(m\) be an even rational integer. Suppose there is a prime \(\ell\) satisfying \(m^ 2\equiv 3\pmod \ell\) and \(e\equiv 0\pmod 3\), where \(e\) is the order of 2 modulo \(\ell\). Set \(a=m(m^ 2 -3)\), \(b= 3m^ 2-1\) and \(c= m^ 2+1\). Further, suppose \(b\) is a prime. In the note under review it is proved that the exponential diophantine equation \(a^ x+ b^ y= c^ z\) has the only positive integral solution \((x,y,z)= (2,2,3)\). Some examples satisfying the above conditions are given. It is not known whether there are infinitely many such \(m\).