AND A BOUND FOR THE FIRST FACTOR OF THE CLASS NUMBER1 L. CARLITZ 1. Let p be a prime e¿3. For (r, p) = l define r' by means of rr' = l(mod£); also let R(r) denote the least positive residue of r (mod p). Maillet defined the determinant Dp of order (p—1)/2 by means of (1) A, R(rs') | (r, s = 1, • • • , (p l)/2) and raised the question whether Dp^0 for all p. Malo computed Dp for p^ 13 and conjec...

The equitable presentation of the quantum algebra $U_q(\widehat{sl_2})$ is considered. This was originally introduced by T. Ito and P. Terwilliger. In this paper, following Terwilliger's recent works (nonstandard) positive part type $U_q^{IT,+}$ its second realization (current algebra) $U_q^{T,+}$ are studied. A for given in terms a K-operator satisfying Freidel-Maillet equation condition on de...