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Some variations and consequences of the Kummer-Mirimanoff congruences. (English) Zbl 0738.11031
Let $$p$$ be an odd prime. It is well known that if the equation $$x^ p+y^ p+z^ p=0$$ is satisfied in non-zero integers $$x$$, $$y$$ and $$z$$ prime to each other and to $$p$$, then $$t\in W=\{-y/x$$, $$-x/y$$, $$-z/y$$, $$- y/z$$, $$-x/z$$, $$-z/x\}$$ is a solution of the Kummer-Mirimanoff congruences, that is, $$\phi_{p-1}(t)\equiv 0\pmod p)$$ and $$B_{2n}\phi_{p-2n}(t)\equiv 0\pmod p$$ $$(1\leq n\leq(p-3)/2)$$, where $$\varphi_ m(X)=\sum^{p-1}_{i=1}i^{m-1}X^ i$$ and $$B_ n$$ is the $$n$$th Bernoulli number defined by $$X/(e^ X-1)=\sum^ \infty_{k=0}(B_ k/k!)X^ k$$.
In this paper several kinds of variations and consequences of the Kummer- Mirimanoff congruences are given. All results are based on the following two relations: let $$S_ m(n)=\sum^ n_{i=1}i^ m$$, $$S_ m'(n)=\sum^ n_{i=1}(-1)^{n-i}i^ m$$ and $$B_ i'=\{(1-2^ i)/2\}B_ i$$. If $$1\leq k\leq p-1$$ and $$m\leq p-3$$, then \medskip\line{(i)    $$k^{p-1-m}\varphi_ p(X)+{p-1-m\over 2}k^{p- 2-m}\varphi_{p-1}(X)+$$\hfil} \medskip\line{\hfil $$+\sum^{p-2- m}_{i=2}{p-1-m\choose i}k^{p-1-m-i}\{B_ i\varphi_{p-i}(X)\}=(p-1- m)\sum^{p-1}_{i=1}i^ mS_{p-2-m}(ik)X^ i,$$} \medskip\line{(ii)    $$(1/2)k^{p-2-m}\varphi_{p-1}(X)-\sum^{p-2- m}_{i=1}{p-2-m\choose i}k^{p-2-m-i}\{B_{i+1}'\varphi_{p-1- i}(X)\}=$$\hfil} \medskip\line{\hfil $$=\sum^{p-1}_{i=1}i^ mS_{p-2- m}'(ik)X^ i-\varphi_{m+1}((-1)^ kX)B_{p-1-m}'.$$}\medskip.
Reviewer: T.Agoh (Tokyo)

##### MSC:
 11D41 Higher degree equations; Fermat’s equation 11A07 Congruences; primitive roots; residue systems 11B68 Bernoulli and Euler numbers and polynomials
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