If a mod m is a unit then aφ(m) ≡ 1 mod m by Euler’s theorem. (When m = p is prime, then this becomes ap−1 ≡ 1 mod p for a 6≡ 0 mod p, which is Fermat’s little theorem.) Depending on a, it might happen that an ≡ 1 mod m for an n that is smaller than φ(m). Example 1.1. Take m = 11. Then a10 ≡ 1 mod 11 for all a 6≡ 0 mod 11. When a = 2 no exponent smaller than 10 works, but when a = 3 we can use ...

In this paper, we prove arithmetic properties modulo 5 and 7 satisfied by the function pod(n) which denotes the number of partitions of n wherein odd parts must be distinct (and even parts are unrestricted). In particular, we prove the following: For all n ≥ 0, pod(135n + 8) ≡ 0 (mod 5), pod(135n + 107) ≡ 0 (mod 5), pod(135n + 116) ≡ 0 (mod 5), pod(675n + 647) ≡ 0 (mod 25), pod(3375n + 1997) ≡ ...

On primitive roots of 1 mod p k , divisors of p ± 1, Wieferich primes, and quadratic analysis mod p Abstract Primitive roots of 1 mod p k (k > 2 and odd prime p) are sought, in cyclic units group G k ≡ A k B k mod p k , coprime to p, of order (p − 1)p k−1. 'Core' subgroup A k has order p − 1 independent of precision k, and 'extension' subgroup B k of all p k−1 residues 1 mod p is generated by p...

Given a nite dimensional algebra over an algebraically closed eld one frequently disregards the projective objects in the category mod of nite dimensional-modules and focuses on the stable category mod. The objects of mod are those of mod but the Hom-groups are the ordinary Hom-groups modulo the subgroup of those morphisms which factor through a projective-module. In this paper we show that mod...

A 3 copies of the manuscript A please indicate the corresponding author's details on the manuscript (tel/fax/e-mail) A 3 sets of figures A electronic version of the manuscript (Label the floppy disk with the type of computer and word processing package used, your name and the name of the file on the disk.) A electronic version of the figures For Review Papers: A 3 copies of the manuscript A ple...

A 4 copies of the manuscript A please indicate the corresponding author's details on the manuscript (tel/fax/e-mail) A 4 sets of figures A electronic version of the manuscript (Label the floppy disk with the type of computer and word processing package used, your name and the name of the file on the disk.) A electronic version of the figures (optional, see instructions below) For Review Papers:...

A 3 copies of the manuscript A please indicate the corresponding author's details on the manuscript (tel/fax/e-mail) A 3 sets of figures A electronic version of the manuscript (Label the floppy disk with the type of computer and word processing package used, your name and the name of the file on the disk.) A electronic version of the figures For Review Papers: A 3 copies of the manuscript A ple...

A 3 copies of the manuscript A please indicate the corresponding author's details on the manuscript (tel/fax/e-mail) A 3 sets of figures A electronic version of the manuscript (Label the floppy disk with the type of computer and word processing package used, your name and the name of the file on the disk.) A electronic version of the figures For Review Papers: A 3 copies of the manuscript A ple...