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Note on the congruences $$2^{p-1}\equiv 1\pmod {p^2}$$, $$3^{p-1}\equiv 1\pmod {p^2}$$, $$5^{p-1}\equiv 1 \pmod {p^2}$$. (English) Zbl 1024.11002
For a prime $$p>5$$ let $$H_0$$ be the subgroup of index $$p$$ of the group of reduced residue classes mod $${p^2}$$ and let $$H_1,\dots ,H_{p-1}$$ be the corresponding cosets. It is proved that if $$p$$ satisfies a rather complicated condition (which can be interpreted as a statement about unit signatures in a cyclotomic field and which holds for all primes below $$50 000$$), then the Wieferich congruence $$2^{p-1}\equiv 1\pmod {p^2}$$ holds if and only if for every $$i$$ the numbers $\sum _{a\in H_i, a<p^2/2}a,\quad \sum _{a\in H_i, p^2/4<a<p^2/2}1$ and $$(p^2-1)/8$$ are of the same parity. A similar result is also proved for the congruences $$3^{p-1}\equiv 1\pmod {p^2}$$ and $$5^{p-1}\equiv 1\pmod {p^2}$$.

##### MSC:
 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 11A07 Congruences; primitive roots; residue systems 11R18 Cyclotomic extensions
##### Keywords:
Wieferich congruence
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##### References:
 [1] Z. I. Borevič I. R. Šafarevič: Teorija čisel. Nauka, Moskva, 1972. [2] W. Narkiewicz: Elementary and analytic theory of algebraic numbers. Polish Scientific publisher, Warszawa and Springer Verlag Heidelberg, 1990. · Zbl 0717.11045
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