×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: EuDML
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.