On some congruences concerning the criteria of Kummer (Q788012)

Suppose that, for an odd prime \(p\), the Fermat equation \(x^ p+y^ p+z^ p=0\) has a solution \((x,y,z)\) in the first case. Then, by a well known result of Mirimanoff, \(B_{p-n}\cdot P_ n(t)\equiv 0\pmod p\) for \(n=3,5,\ldots,p-2\), where \(B_ 2\), \(B_ 4,\ldots\). are Bernoulli numbers, \(P_ n(X)\) are certain polynomials (that do not depend upon \(p\)) and \(t\) is any one of the six numbers \(x/y,\ldots,y/z\). This leads to the result that \(B_{p-n}\equiv 0\pmod p\), if certain two integers derived from \(P_ n(X)\) and \(t\) are not divisible by \(p\). The author proves that the two numbers in question vanish mod \(p\) for all \(n>1\), \(n\equiv 1\pmod {p-1}\), excluding a few small values of \(p\).