Number theoretic consequences on the parameters of a symmetric design (Q1821103)

In studying symmetric designs one meets the diophantine equation (*) \(k(k-1)=\lambda (v-1)\). The author shows by direct application of known results from diophantine equations that, keeping \(\lambda\) fixed, (*) has finitely many solutions if either i) v consists of a fixed set of primes, or ii) v is an n-th power (n\(\geq 3\) fixed) or iii) \(v\lambda\) is a square.