chore: update readme with new todos and features

README.md

@@ -1,8 +1,8 @@
 # primality
 
-Primality is a Golang library for determining whether any given integer $i\in
-\mathbb{Z}^+,~i\ge1$. The library provides two implementations of algorithms for
-determining the primality of an arbitrarily sized integer, using the `math/big`
+Primality is a Golang library for determining whether any given integer
+$i\in\mathbb{Z}^+,~i\ge1$. The library provides two implementations of algorithms
+for determining the primality of an arbitrarily sized integer, using the `math/big`
 library.
 
 The two methods provided are the Miller-Rabin and AKS primality test the former
@@ -16,20 +16,25 @@ integers.
 ### Miller-Rabin:
 
 - Highly accurate probabilistic primality test
-  - 25 rounds and force usage of base 2 recommended
+  - 25 rounds and force usage of base 2 recommended for near 100% accuracy
 - Arbitrarily large integer support (`big.Int`)
+- $\mathcal{O}(r\cdot s)$ time complexity (assuming `big.Int` operations are
+  $\mathcal{O}(1)$ where $r$ is the number of repetitions and $s$ the number of
+  trailing zeros on $n$, the number being tested - otherwise it is related to
+  the operations of `big.Int` integers and $n$ itself.
+  - As a prime must be odd $s\le7$ meaning the time complexity in its worst case
+    is $\mathcal{O}(175)=\mathcal{O}(1)$ with 25 repetitions.
 
 ### AKS
 
 - Deterministic primality test
-  - Slow on larger integers
+  - Slow overall - but guarantees 100% a valid outcome
 - `int` support only
 
 ## TODOs
 
-- Fix false positives with AKS method
-- Add unit tests
 - Improve speed of the AKS method
-  - Specifically step 5
+  - Specifically step 5 but overall it is slow
 - Make the AKS method work on arbitrarily sized integers
   - use `big.Int`s over `int`s
+- Determine the true time and space complexities for both methods