On the first case of Fermat's last theorem. II (Q1074621)

Let p be an odd prime and consider the equation (*) \(x^ p+y^ p+z^ p=0\) in the first case (i.e., \(p\nmid xyz),\) where x, y, z are nonzero integers prime to each other. As the criterion of (*) in the first case, it is well-known that the Kummer congruences \([d^{p-2}\{U_ t(v)\}/dv^{p-2}]_{v=0}\equiv 0\quad (mod p)\) and \(B_{2k}[d^{p-1- 2k}\{U_ t(v)\}/dv^{p-1-2k}]_{v=0}\equiv 0\quad (mod p)\) \((k=1,2,...,(p-3)/2)\) hold, where \(t\in \{-y/x\), -x/y, -z/y, -y/z, -x/z, -z/x\(\}\), \(U_ t(v)=1/(1-te^ v)\) and \(B_{2k}\) is the Bernoulli number with the even index notation. By Mirimanoff's results [see p. 139- 148 in \textit{P. Ribenboim}, 13 Lectures on Fermat's Last Theorem (1979; Zbl 0456.10006)] one can easily see that if \(t\not\equiv 0\), 1 (mod p), then the above congruences are equivalent to \([d^{p-2}\{U_ t(v)\}/dv^{p-2}]_{v=0}\equiv 0\quad (mod p)\) and \[ [d^ k\{U_ t(v)\}/dv^ k]_{v=0}\quad [d^{p-2-k}\{U_ t(v)\}/dv^{p-2- k}]_{v=0}\equiv 0\quad (mod p) \] \((k=1,2,...,(p-3)/2).\) In this paper, the author discusses the p-divisibility properties of the numerators of \(B_ k\) and \([d^ k\{U_ t(v)\}/dv^ k]_{v=0},\) and derives some consequences from the above congruences.