### Exercises 1. *(For mathematicians only.)* Let *p* be a (large) prime, and let *a* be a primitive root ***modulo*** *p*. Then if *b*ᵢ is the residue of *a*ᵢ ***modulo*** *p* (1 ≤ *b*ᵢ ≤ *p*-1 ), the sequence: *b*ᵢ-1\ *p*-1 is a cyclical sequence of *p*-1 distinct numbers in the range **0** to **1** (excluding **1**). By choosing *a* suitably, these can be made to look fairly random. **65537** is a Fermat prime, **2¹⁶+1**. Because the multiplicative group of non-zero residues ***modulo*** **65537** has a power of **2** as its order, a residue is a primitive root if and only if it is not a quadratic residue. Use Gauss' law of quadratic reciprocity to show that **75** is a primitive root ***modulo*** **65537** . The *ZX Spectrum Next* uses *p*=**65537** and *a*=**75**, and stores some *b*ᵢ-1 in memory. **RND** entails replacing *b*ᵢ-1 in memory by *b*ᵢ₊₁-1, and yielding the result (*b*ᵢ₊₁-1) / (*p*-1). **RANDOMIZE n** (with **1** ≤ **n** ≤ **65535**) makes *b*ᵢ equal to *n*+1. **RND** is approximately uniformly distributed over the range **0** to **1**.