1 Division 3 1.1 Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Greatest common divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Fundamental theorem of arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . ....

A prime number is an integer bigger than l that has no factor except l and itself . A number that is not prime is called composite. The primality testing problem is to decide whether a given integer is prime or composite. It is considered to be well solved, in contrast to ine factoring problem, which asks for the factorization of a given integer into prime factors. As an example of what can be ...

The topic of my talk is number theory in geometry. I will say a few words about history of mathematics to show how geometry has played an important role in the development of mathematics. I will also mention some contributions of Indian Mathematicians of antiquity since these are not widely known. Humans, and apparently some animals too, can and need to count. We count discrete objects, such as...

Multiplicative Number Theory 638 27.2 Functions . . . . . . . . . . . . . . . . . 638 27.3 Multiplicative Properties . . . . . . . . . 640 27.4 Euler Products and Dirichlet Series . . . 640 27.5 Inversion Formulas . . . . . . . . . . . . 641 27.6 Divisor Sums . . . . . . . . . . . . . . . 641 27.7 Lambert Series as Generating Functions . 641 27.8 Dirichlet Characters . . . . . . . . . . . . 642...

Background. The integer distance graph G(Z, D) with distance set D = {d1, d2, . . .} has the set of integers Z as the vertex set and two vertices x, y ∈ Z are adjacent if and only if |x − y| ∈ D. The integer distance graphs (under Euclidean norm) were first systematically studied by Eggleton–Erdős–Skilton in 1985 [12, 13], and have been investigated in many ways [50, 56, 57, 61]. One of main go...

These are the notes for a course taught at the University of Michigan in F92 as Math 676. They are available at www.math.lsa.umich.edu/∼jmilne/. Please send comments and corrections to me at [email protected]. v2.01 (August 14, 1996.) First version on the web. v2.10 (August 31, 1998.) Fixed many minor errors; added exercises and index.

D oes the equation x + y + z = 29 have a solution in integers? Yes: (3,1,1), for instance. How about the equation x + y + z = 30? Again yes, although this was not known until 1999: the smallest solution is (−283059965,−2218888517, 2220422932). And how about x + y + z = 33? This is an unsolved problem. Of course, number theory does not end with the study of cubic equations in three variables: on...

2 Arithmetic in an algebraic number field 30 2.1 Finitely generated abelian groups . . . . . . . . . . . . . . . . . . 30 2.2 The splitting field and the discriminant . . . . . . . . . . . . . . 31 2.3 Number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4 The ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.5 Ideals . . . . . . . . . . . . . . ....