This seems simple enough, but let’s play with this definition. The Pythagoreans, an ancient sect of mathematical mystics, said that a number is perfect if it equals the sum of its positive integral divisors, excluding itself. For example, 6 = 1 + 2 + 3and 28 = 1 + 2 + 4 + 7 + 14are perfect numbers. On the other hand, 10 is not perfect because 1 + 2 + 5 = 8, and 12 is not perfect because 1+2+3+4...

We give an informal introduction to the most basic techniques used to evaluate moments on the critical line of the Riemann zeta-function and to find asymptotics for sums of arithmetic functions.

The focus of this entire note is on positive integers. I will start by the basic notion of divisibility. We say that a divides b, or a is a divisor of b, or b is a multiple of a, if there exists an integer m such that b = am We also denote this by a | b (a divides b). If a is not a divisor of b, the we write a | b, but we can still talk about division by a with a remainder r. Given any two inte...

The book is a selection of papers from a wide variety of topics in Cryptography and Number Theory, presented at the 33rd Annual Meeting of the Australian Mathematical Society and at the Workshop on Number Theory and Cryptography in Telecommunications in 1989. The book is composed of three parts that focus on a range of topics such as stream ciphers, applications of cryptography, number theory, ...

The division relation plays a prominent role throughout this note, and so, we start by presenting some of its basic properties and their relation to addition and multiplication. First, it is re exive because multiplication has a unit (i.e., m= 1×m) and it is transitive, since multiplication is associative. It is also (almost) preserved by linear combination because multiplication distributes ov...