A set of \(m+1\) integers \(k_0, k_1,\dots,k_m\) with the property that the \(m^2+m\) differences \(k_i-k_j\) \((i\neq j\), \(i, j=1, 2, \dots, m)\) are congruent modulo \(q\), \(q = m^2 + m + 1\), to the integers \(1, 2, \dots, m^2+m\) in some order is called a perfect difference set. A number \(t\) is a multiplier of a set \(k_0, k_1,\dots,k_m\) if the numbers \(tk_0, tk_1,\dots, tk_m\) are congruent modulo \(q\) to the integers \(k_0+s, k_1+s, \dots, k_m+s\) for some \(s\). \textit{J. Singer} [Trans. Am. Math. Soc. 43, 377--385 (1938; Zbl 0019.00502)] has proved, by using the method of projective geometry over finite fields, that a sufficient condition for the existence of a perfect difference set of \(m+1\) integers is that \(m\) is a power of a prime. He also conjectures that \(t\) is a multiplier of a perfect difference set of \(p^n+1\) integers if and only if \(t\) is congruent to a power of \(p \pmod q\). The author proves Singer's theorem by means of the method of the elementary theory of finite fields and he shows that the hypothesis on a multiplier is valid for special perfect difference sets.