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On the Kummer-Mirimanoff congruences. (English) Zbl 0648.10013
Let p be an odd prime, \(B_ m\) the m-th Bernoulli number defined by \(v/(e^ v-1)=\sum ^{\infty}_{k=0}(B_ k/k!)v^ k\) and \(\phi _ n(v)=\sum ^{p-1}_{i=1}i^{n-1}v^ i\) for any integer n. In 1857 Kummer showed that if \(\nu\) and \(\rho\) are units in \({\mathbb{Z}}_ p\) (the ring of all rational numbers which are p-integral) such that \(\nu ^ p+\rho ^ p-1=0,\) then \(B_ m\phi _{p-m}(t)\equiv 0 (mod p)\) \((m=1,2,...,p-3)\) hold for \(t=\nu,\rho\). On the other hand, Mirimanoff proved that Kummer’s congruences hold for \(t=t'\not\equiv 0,1 (mod p)\) if and only if \(\phi _ m(t)\phi _{p-m}(t)\equiv 0 (mod p)\) \((m=1,2,...,(p-1)/2)\) hold for \(t=t'\). A relation between \(B_ j\phi _{p-j}(t)\) and \(\phi _ i(t)\phi _{p-i}(t)\) may be given by the congruence \[ \frac{1+t}{2}\phi _{p-1}(t)+\frac{1-t}{m+1}\sum ^{p-2- m}_{j=2}\left( \begin{matrix} p-1-m\\ j\end{matrix} \right)B_ j\phi _{p- j}(t)\equiv -\sum ^{m+1}_{i=2}\left( \begin{matrix} m\\ i-1\end{matrix} \right)\phi _ i(t)\phi _{p-i}(t)(mod p), \] where \(t\not\equiv 0,1 (mod p)\) and m is any integer with \(1\leq m\leq p-4.\)
The main purpose of this paper is to derive some systems of congruences which have the same solutions as those of Kummer-Mirimanoff’s congruences. Furthermore, the first case of Fermat’s last theorem is discussed.
Reviewer: T.Agoh

11D41 Higher degree equations; Fermat’s equation
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11A07 Congruences; primitive roots; residue systems
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