Combinatorial Proofs of Fermat's, Lucas's, and Wilson's Theorems

Abstract

(L(p)[Q], Q)k = (PQ, PQ)k+m = +IIell2m |> IIPII\QII, (5) showing that L(p) is positive. Therefore, we see that all eigenvalues of L(p) are positive and, on the basis of (5), that II P II| furnishes a lower bound for them. Furthermore, equality holds in (4) if and only if either P 0 or II P II| is the smallest eigenvalue of L(p) and Q is an eigenvector corresponding to it (unless Q = 0). A particular case in which equality holds in (4) occurs when P = P(y) belongs to Nm and Q = Q(z) to Nk, where y E RP, z E Rq, and IR = RP x Rq. Added in proof. Professor Luo Xuebo, who was one of his coauthor's Ph.D. supervisors, died in March 2004. Zhu-Jun Zheng expresses his deep respect for and everlasting memory of his deceased colleague and mentor.