2
votes
1answer
377 views

Find point on line that has integer coordinates

Given a 2D line e.g. (3,10) -> (8.3,16.5), how can I find any point on that line that has has whole-number coordinates? I can easily iteratively walk along one ...
2
votes
1answer
59 views

Lattices in $\mathbb{C}$

I have the following assignment: consider the map $$|\cdot|:\mathbb{Z}[i]\longrightarrow \mathbb{N},\qquad |a+ib|:=a^2+b^2$$ 1) Prove that $|\alpha|<|\beta|$ iff $|\alpha|\leq |\beta|-1$ and ...
0
votes
2answers
80 views

Relating lattice bases in $\mathbb R^2$

this is a homework question but I am pretty confused on it--just don't know where to start. We're given a lattice basis $(a, b)$ for a lattice $L$ in $\mathbb{R}^2$, and are supposed to show that ...
1
vote
1answer
660 views

Lattice Paths to $(2n, 2n)$ that do not go through $(n,n)$

I know that from the origin to $(x,y)$ there are $${x+y \choose x} = {x+y \choose y}$$ Lattice paths. The question is finding the number of paths from the origin to $(2n, 2n)$ not passing through ...