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:

This equation defines a plane in 3D space
All RANDU outputs lie on just 15 such parallel planes
This makes RANDU catastrophically bad for Monte Carlo simulations
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.

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

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