p-adic numbers in a video

[Olli]

Small movie about p-adic numbers :
https://m.youtube.com/watch?v=3gyHKCDq1YA#

[jack]

the graphics look intriguing, but what are p-adic numbers good for?
I asked ChatGPT: are there practical applications for p-adic numbers?

Yes, **p-adic numbers** have several practical applications across various fields, despite their abstract mathematical origin. Some of the key areas where p-adic numbers are used include:

### 1. **Cryptography**:
- **Elliptic Curve Cryptography**: p-adic numbers are sometimes used in computational number theory, especially in algorithms that deal with elliptic curves, which are important in modern cryptography (e.g., public-key cryptography).
- **p-adic L-functions**: In cryptographic protocols, certain p-adic analytic functions can play a role in creating cryptographically secure algorithms.

### 2. **Mathematical Physics**:
- **Quantum Physics**: p-adic numbers have been proposed as a framework for certain quantum systems and string theory models. For instance, p-adic analysis offers an alternative to real-number-based models in describing space-time or quantum fields.
- **String Theory**: p-adic numbers have been applied in the study of non-Archimedean geometries, which can provide new insights into string theory, particularly in the study of p-adic string theory.

### 3. **Biological and Cognitive Modeling**:
- **p-adic models of cognition**: In the field of cognitive science, p-adic numbers have been used to model hierarchical structures such as memory or cognitive processes. The hierarchical nature of p-adic numbers makes them suitable for representing tree-like structures or nested systems, which occur frequently in cognition and neural networks.

### 4. **Computer Science and Data Structures**:
- **Fractal and Tree Structures**: p-adic numbers are used in the study of data structures with recursive, hierarchical properties, such as trees. Their non-Archimedean geometry makes them particularly effective in analyzing fractals, hierarchical clustering, and data structures with multiple layers.

### 5. **Number Theory and Algebraic Geometry**:
- **Diophantine Equations**: In number theory, p-adic numbers provide powerful tools to solve Diophantine equations. p-adic methods, such as Hensel's lemma, are crucial in the study of rational solutions to polynomial equations.
- **Local-Global Principle**: The p-adic numbers are used in proving theorems related to the local-global principle, where local solutions to equations (modulo powers of primes) are used to infer global solutions.

### 6. **Analysis and Differential Equations**:
- **p-adic Dynamical Systems**: p-adic numbers are used to study dynamics and behaviors of systems that evolve according to recursive rules. They are especially effective in studying systems with discrete scaling, where the real number system may not offer the right tools.
- **p-adic Differential Equations**: p-adic analysis can be applied to solve certain types of differential equations, particularly those that arise in number-theoretic contexts or involve discrete symmetries.

### 7. **Economics and Game Theory**:
- **p-adic Utility Functions**: p-adic numbers have been explored as a potential tool to model preferences and utility functions in economic theory, particularly in situations involving hierarchical choices or decisions.

These examples show how p-adic numbers, despite their seemingly abstract nature, can be very useful in applied fields ranging from cryptography to quantum physics and beyond.

[Olli]

Your IA is very impressive, in the way it gives seven several domains, what it is really pedagogic.

But it misses the numeric rates of tool uses, which are replaced by a litteral description. However this litteral description is very rich, and offers to a non-englo as me, to discover a rich and strategic vocabulary, it misses the values of these rates.

ex : << it is strongly recommanded >> => how many percent in a total of way.
By the way, which unit ? (= which scale?)

(little scale tendancy)
percent ? part per million ? part per billion ?

(huge scale tendancy)
percent ? number size ? number size of the number size ?

And as it misses numeric details, it also misses the sources, the dates of the discoveries, the authors, the institutes in which these discoveries have been allowed, and the communication support (papers, sites, etc...)

My synthesis, after this read, after a own long period away, is this IA is very useful :
- to structure globally a future document
- to measure the approximative duration to search, verify and write a future document
- to keep a global scope of the unfinished work. So, to adapt the work load to the rythm of the researcher, student, enginner.

So it is very impressive. Love Euclid, and thank you jack.

[Olli]

[...] of non-Archimedean geometries [...]

Apparently, seems to be outside these numbers domains (and future hypothetic numbers sets) :