Sage for Newbies

Major Goals : Sage Primers

Basics

• Primer Guidelines primer_template\example.sws

• Primer Design Principles primer_design_principles.rtf

• SAGE as a Smart Calculator (target: Freshmen) sage_as_a_smart_calculator.sws

Calculus

• Differential Calculus (target: Freshmen) differential_calculus.sws

• Integral Calculus (target: Freshmen)

Number Theory

• Quadratic Forms (target: Arizona Winter School Participants) quadratic_forms.sws

• Number Theory via Diophantine Equations (target: Elementary Number Theory students)

• Number Theory via Primes (target: Elementary Number Theory students) number_theory.primes_0.1.sws

Abstract Algebra

• Group Theory by Robert Beezer (target: Undergraduate Math Majors) group_theory.sws

Target

1) Accessible to high school math teachers and undergraduate mathematics majors.

2) Anticipated user desires

a. Content specific modules

i. Quadratic Forms

ii. Group theory

iii. Abstract algebra

iv. Calculus

v. Number theory

vi. High school algebra / trigonometry / precalculus

vii. Probability

viii. Statistics

b. Plotting 2 and 3 dimensions

c. Sage math functions (sage as calculator), sage constants

d. Generate Classroom examples

i. show (), latex()

ii. matplotlab

3) Demonstrate SAGE functionality:

a. Primes

b. Random numbers

c. Plotting

d. Interact

e. Sage data types

4) Programming

a. Types, casting, relevant Sage data types

b. Lists, tuples

c. Control operators (if, then, else, logical operators, in, srange())

d. Loops

i. For, in, srange(), range()

e. Functions

f. Recursion

5) Topics

a. Primes and factorization

i. Given a random number, is it a prime?

1. Modular division

a. random()

b. Factor()

2. Euclidean algorithm

a. Recursion

b. gcd()

3. primality testing

a. for loops

b. range()

c. is_prime()

ii. How many primes are there?

1. prime_pi()

2. plotting example

iii. Where are the primes?

1. Density of primes

2. primes()

3. Arithemtic sequences of primes

b. Diophantine equations

i. Linear Diophantine equation

1. extended euclidean algorithm

2. recursion vs iteration

ii. diagonal quadratic forms; sums of squares (ENT p. 25)

1. Pythagorean triples and generating them

2. Graphing the Pythagorean triples

3. Enumerating all triples using linear intersections

4. Elliptic curves and congruent numbers (chapter 6, stein)

iii. Pell’s Equation (?)