[Hint] : Assume 3 cases, then by Elimination using Mr Brown’s hint (somethingS right, somethingS wrong)
Fermat Little Theorem: For any prime integer p, any integer m
When m = 2,
Note: 九章算数 Fermat Little Theorem (m=2)
Pascal Triangle (1653 AD France ）= (杨辉三角 1238 AD – 1298 AD)
1 4 6 4 1 => sum = 16= 2^4 (4 is non-prime)
What is your favorite theorem ?
I have 2 theorems which trigger my love of Math :
- Chinese Remainder Theorem: 韩信点兵, named after a 200 BCE Han dynasty genius general Han Xin （韩信） who applied this modern “Modular Arithmetic” in battle fields.
- Fermat’s Last Theorem：The Math “prank” initiated by the 17CE amateur Mathematician Pierre de Fermat kept the world busy for 380 years until 1974 proved by Andrew Wiles.
Note 1: Lycée Pierre de Fermat (Classe Préparatoire) happens to be my alma mater named after this great Mathematician born in the same southern France “Airbus City” Toulouse.
Note 2: His another Fermat’s Little Theorem is used in modern computer cryptography.
Geometry and Computing Math
– Prof ST Yau (Harvard University Tenured Professor, Fields Medalist 1982, Wolf Prize 2010)
AI must be supported by solid Math Theory for it to be fully further developed.
This statement truly reflects the bottle-neck faced by the AI 2.0 (Expert Systems) in the 1980s using a non-rigorous “Fuzzy Logic” Math.
Current AI 3.0 (Deep Learning) is using Calculus (Cauchy Gradient Descent) to compute, it is empirical and sans proven math theoretical support.
The new Math tools like Persistent Homology (持续同调论) , Comformal Geometry 共形几何, etc may be the answer for future AI 4.0.
1. 蒙日-安培方程 Mongo-Ampere Equation
2. 共形(保角) 映射 Comformal Mapping
3. 仿射几何 Affine Geometry
4. 持续同调论 Persistent Homology
5. 叶状结构 Foliation Structure