A prime is called elite, or anti-elite, when all but finitely many Fermat numbers are quadratic nonresidues or residues, respectively, modulo . It is known that if the multiplicative order of 2… Expand

A prime p is elite if all sufficiently large Fermat numbers Fn = 22n + 1 are quadratic nonresidues modulo p. In contrast, p is anti-elite if all sufficiently large Fn are quadratic residues modulo p.… Expand

A primitive root g modulo n is when the congruence gx ≡ 1 (mod n) holds if x = φ(n) but not if 0 < x < φ(n), where φ(n) is the Euler’s function. The primitive root theorem identifies all the positive… Expand

These notes are prepared for the students at Philadelphia University in Jordan who are taking the Math 342–442 series of Abstract Algebra. Topics in group theory are covered in the first thirteen… Expand

Using prime numbers whose digits are zeros and ones, we demonstrate how to construct integers $m$ for which $mP$ is a Smith number for any prime $P$ with a fixed, small digital sum. Conversely, using… Expand

We detail the proof of the fundamental theorem of finite abelian groups, which states that every finite abelian group is isomorphic to the direct product of a unique collection of cyclic groups of… Expand

We show that the sequencew k mod n, given that gcd(w;n) > 1, can reach a maximal cycle length of (n) if and only ifn is twice an odd prime power,w is even, andw is a primitive root modulon=2.