On Primitive Singular Numbers

Recently there are tremendous interest in finding primitive singularnumber due to applications in number theory and in integral matrix theory. In [10],Hong, Shum and Sun proved Hong's claim saying that 180 is the least primitive singularnumber. In this paper, we prove another claim of Hong stating that 270 is the secondleast primitive singular number. We also show that if S is a gcd-closed set satisfyingxi < 270 and xi = 180 for all 1≤i≤n , then the LCM matrix on S is nonsingular. Wealso study the primitive singular number of the form 2pqr, and find out all primitivesingular numbers of this form such that p,q,r not exceeding 1000.