# Why Prime Numbers Still A Mistery

Prime Numbers since Euclid 2300 years ago remains a mystery today…

Riemann’s Hypothesis is the Top Conjecture Unsolved in 21st Century.

https://theconversation.com/why-prime-numbers-still-fascinate-mathematicians-2-300-years-later-92484

# Prime Number Theorems Conjectures Explained

1. Twin Primes Conjecture : there are infinitely many pairs of Twin Primes among the infinitely many prime numbers.

Note: 2013 Zhang YiTang proved the gap between the twin primes is no more than 70,000,000 [Terence Tao’s Polymath Project using Zhang’s method to further narrow the gap to 246 ]

Chen’s “1+2” Theorem

3. Palindromic Primes (eg.11, 101, 16561)

# BM Category Theory II 8: F-Algebra, Lambek’s Lemma , Catamorphism, Coalgebra, Anamorphism

[Continued from previous BM Category Theory …]

$\boxed { \text {type Algebra f a = f a} \to \text {a} }$

Intuition: [Artificial Intelligence] You teach the computer like to a Primary 6 kid, that Algebra is a type of expression (f) which, after evaluation,  returns a value.

If a = i (initial) [or u (terminal)],
$\boxed { \text {(f i} \to \text {i )} \implies \text {f = Fix-point} }$

Intuition: Fix-point because, the Initial “i”, after evaluating the expression f, returns itself “i”.

Lambek’s Lemma
$\boxed { \text {Initial Algebra is an Isomorphism} }$

Note: Endo-functor is a functor (equivalent to function in Set Theory) within the same Category (Endo = Self = 自)

Video 8.1 F-Algebras & Lambek’s Lemma

Video 8.2 Catamorphism & Anamorphism

foldr ~ catamorphism (浅层变质) of a Fix-point endo-functor on a List.

Examples: Fibonacci, Sum_List

Remark: Cool Math! the more  advanced concept it is, the more closer to Nature (eg.Geology, Biology) : Catamorphism 浅层(风化)变质, or “thin-layer change in nature” (in Functional Programming languages: foldr or map) eg : add1 to a list (1 5 3 8…)
= (2 6 4 9 …)

$\boxed { \text {type Coalgebra f a = a} \to \text {f a} }$

Intuition: Reverse of Algebra, given a value, Coalgebra returns an expression (f).

Anamorphism (合成变质) ~ unfoldr

Example: Prime numbers

Remark: Anamorphism (合成变质) or “synthesised change in nature“: eg. Start from a  “seed” prime number “2” generates  all other infinite prime numbers (3 5 7 9 11 13 17 …)

Note: In Haskell, no difference between Initial and Terminal Fix-points. However, since Fix-point is not unique, in Category Theory there is the Least Fix-point (Initial) and Greatest Fix-point (Terminal).

Ref:

Reading “Understanding F-Algebra ” by BM: https://bartoszmilewski.com/2013/06/10/understanding-f-algebras/