Chapter 4 Definitions

Question 1

A Diaphantine Equation

Answer 1

A Diaphantine Equation is an equation in one or more variables for which integer solutions are sought.

Question 2

A lattice in Zⁿ

Answer 2

A lattice in Zⁿ is a subgroup L ⊆ Zⁿ of finite index.

Question 3

1. Symmetric Subset
2. Convex Subset

Answer 3

A subset S ⊆ |Rⁿ is:

1. Symmetric if x ϵ S ⇒ -x ϵ S
2. Convex if x, y ϵ S ⇒ tx + (1-t)x ϵ S
   for all t with 0 ≤ t ≤ 1

Question 4

Volume of a ball of radius r in 4-space is:

Answer 4

π²r⁴ / 2

Question 5

Legendre’s Equation

Answer 5

ax² + by² = cz²

Question 6

A primitive solution

Answer 6

A solution will be called primitive if gcd(x,y,z)=1

Question 7

Pythagorean Triple

Answer 7

x = u² - v²

y = 2 u v

z = u² + v²

Where u,v are coprime, u > v and u ≠ v mod 2.