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The Kurepa-Vandermonde matrices arising from Kurepa’s left factorial hypothesis

Meštrović Romeo (University of Montenegro, Maritime Faculty, Dobrota, Kotor, Montenegro)

Kurepa’s (left factorial) hypothesis asserts that for each integer n ≥ 2 the greatest common divisor of !n := Pn-1∑k=0 k! and n! is 2. It is known that Kurepa’s hypothesis is equivalent to p-1∑k=0 (-1)k/k!≡/ 0 (mod p) for each odd prime p, or equivalently, Sp-1 ≡/ 0(modp) (i.e., Bp-1 ≡/1(modp)) for each odd prime p, where Sp-1 and Bp-1 are the (p-1)th derangement number and the (p-1)th Bell number, respectively. Motivated by these two reformulations of Kurepa’s hypothesis and a congruence involving the Bell numbers and the derangement numbers established by Z.-W. Sun and D. Zagier [28, Theorem 1.1], here we give two “matrix” formulations of Kurepa’s hypothesis over the field Fp, where p is any odd prime. The matrices Vp and Cp which are involved in these “matrix” formulations of Kurepa’s hypothesis are the square (p-1)x(p-1) Vandermondelike matrices. Accordingly, Vp and Cp are called the Kurepa-Vandermonde matrices. Furthermore, for each odd prime p we determine det(Vp) and det(Cp) in the field Fp.

Keywords: Left factorial function, Kurepa’s hypothesis, derangement number, reformulation of Kurepa’s hypothesis, Bell number, Kurepa’s determinant, congruence modulo a prime, Kurepa-Vandermonde matrix, Kurepa-Vandermonde determinant