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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 \(5^{p-1}\equiv 1\pmod {p^2}\).

11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11A07 Congruences; primitive roots; residue systems
11R18 Cyclotomic extensions
Full Text: EuDML
[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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