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I'm going to use these old SAGE notes for most of this, but here is my plan and a FLAG:

NOTES ON CLOUD.SAGEMATH.COM (now cocalc)

- What's a lattice?
- GCD is actually mini lattice reduction
- The Knapsack Problem with LLL
- Minimal Polynomials

It's all integer combinations of some vectors. It looks like this:

To play with one we think of vectors and start adding them up.

**Find Me:** Let \(v_1 = (-502, 469, -111)\) and \(v_2 = (101, -95, 23)\)
what is the smallest integer combination of those two you can find?

**Find Me:** Let \(v_1 = (1, 0, 3920)\) and \(v_2 = (0, 1, 4200)\)
what is the smallest integer combination of those two you can find?

Make a matrix in Sage and call `M.LLL()`

Retry those two problems using LLL.