# LaTeX Writeup Notes

## Claude Aside - RANDU Mathematical Verification

**Context:** During development, Claude added a mathematical verification to prove RANDU's planar structure beyond just visual inspection.

**The Insight:**
RANDU's fatal flaw isn't just visible in 3D plots - it can be mathematically proven. Every triplet of consecutive values (x_n, x_{n+1}, x_{n+2}) satisfies the linear relationship:

```
x_{n+2} ≡ 6·x_{n+1} - 9·x_n (mod 2^31)
```

**Why This Matters:**
1. This equation defines a plane in 3D space
2. All RANDU outputs lie on just 15 such parallel planes
3. This makes RANDU catastrophically bad for Monte Carlo simulations
4. The verification code (see problem1.rs:92-109) proves this relationship holds for every triplet

**For LaTeX Writeup:**
Consider adding this mathematical verification as a proof/aside that:
- Shows the limitation isn't just visual - it's algebraically constrained
- Demonstrates that statistical tests (mean, std dev) alone miss critical structural flaws
- Explains *why* RANDU fails: deterministic linear constraint reduces 3D space to 15 planes

**Cool Factor:**
This is a case where the AI assistant didn't just visualize the problem - it provided mathematical proof of the underlying structural flaw. The self-checking approach (generating triplets and verifying the constraint) demonstrates both the problem and a rigorous testing methodology.

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## Other Sections to Include

### Problem 1A - Basic LCG Implementation
- Show first 5 random numbers
- Discuss LCG formula: x_{n+1} = (a·x_n + c) mod m

### Problem 1B - Comparison
- Good LCG vs RANDU
- Statistics comparison (both look fine!)
- 3D scatter plots (visual difference)
- Mathematical verification (definitive proof)

### Figures to Include
- `good_lcg_3d.png` - Uniform 3D distribution
- `randu_3d.png` - Visible planar structure
- Consider adding histogram comparison if needed