# Maths
## Radon transform
converts densities in angular coordinate space to cartisian coordinate space.

## Millenium problems
1. Riemann's hypothesis
2. P vs NP
3. Navier-stokes
4. Hodge Conjecture
5. Yang-Mills and Mass Gap
6. Birch and Swinnerton-Dyer Conjecture
7. Poincare Conjecture

- prime: that has only 1 as factor
- coprimes: those have only 1 as common factor i.e gcd=1
  eg., 6(2x3), 11, 35(5x7),
- twinprimes: those primes whose difference is 2: eg., 3,5

## asymmetric cryptography
### RSA - Revist Shamir Adleman
### DSA - Digital Signature Algorithm
### EC DSA - Ecliptic Curve DSA
### EC DH - Ecliptic Curve Diffie-Hellman

## Euclid's division to find HCF/GCD of a,b where a>b
```
a=bq+r where q is quotient and r is remainder
b=rq1+r1
r=r1q2+r2
r_(n-2)=r_(n-1)q_n+r_n
if r_n==0 then r_(n-1) is the HCF.
```

## modular arithmatic
### X is modular multiplicative inverse of A in mod m or
```
X is moudlo inverse of A under m
AX%M=1
X should be < M
X exists only if gcd(A,M)=1
```

#### Extended Euclidean Algorithm
Used for computing modular multiplicative inverse

### Chinese Remainder Theorem
```
if
x ≡ a1 mod n1
x ≡ a2 mod n2
then
x ≡ a1n2[n2^−1]_n1 + a2n1[n1^−1]_n2
```

### Fermat's little theorem
`a^b%b = a%b ∋ b ∈ P & a ∈ Z`

### Bezout's identity
`(ax + by = gcd(a, b)) ∋ x, y ∈ Z`

## karatsuba multiplication
- fast multiplication

## The Cauchy-Schwarz Inequality
$$\left( \sum_{k=1}^n a_k b_k \right)^2 \leq \left( \sum_{k=1}^n a_k^2 \right) \left( \sum_{k=1}^n b_k^2 \right)$$

## software libraries
### GMP (GNU Multiple Precision Arithmetic Library)
### OpenBLAS (Basic Linear Algebra Subprograms)

## symbols
∵ - since
∴ - therefore
∃ - there exists
∄ - there dosn't exist
⌈x⌉ - ceil function
: - such that
∋ - where
∈ - belongs to
∉ - does not belong to
≡ - equivalent

## Numbers
- googol: 10^100

## Mathematicians
- Madhava: Pi, calculus - 1400 AD
- Newton: calculus - 04/01/1643 - 31/03/1727
- Diophantus: 200 - 284 AD, Arithmetica, solving algebraic equations
- Kaprekar: 6174 = 7641 - 1467 = Kaprekar's number

## Delauny Triangulation
- a point is adjacent to two other points if it doesn't circumscribe any other
  point.
  
## monty-hall problem
- pick one out of 3, out of remaining 2 when one is eleminated and given the chance to switch then difinitely switch coz actually the remaining 2 are 
  given

## fibonacci series | golden spiral
|   |   |
|---+---|
|   |   |
|   |   |
|   |   |
|   |   |

## Breakthrough Math prize
- $3Mn
## Godel prize given for computer research
- 2025 to Eshan Chattopadhyay, IIT kanpur alumni
  - explicit two-source extractor with polylogarithmic min-entropy
## Fields Medal
- $1Mn
- given every 4years for under 40
## Turing Award