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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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