# a-conjecture-of-mine

An exercise on polyglossy: the same problem solved on multiple languages

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Date:   Fri, 14 Jun 2019 13:14:37 -0300

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# A Conjecture of Mine
An exercise on _polyglossy_. The same problem solved on multiple languages.

-## Problem
-Let  <img src="https://latex.codecogs.com/gif.latex?S:\mathbb{N}&space;\mapsto&space;\mathbb{N}"/> be the sum of the digits of a positive integer.
+## The Problem - Mathematicians Version
+Let <img src="https://latex.codecogs.com/svg.latex?S:&space;\mathbb{N}\rightarrow&space;\mathbb{N}"/> be the sum of the digits of a positive integer in a [_base 10 positional number system_](https://en.wikipedia.org/wiki/Positional_notation). We have:

-Prove that <img src="https://latex.codecogs.com/gif.latex?\forall&space;a,&space;b&space;\in\mathbb{N}&space;:&space;S_{a&space;&plus;&space;b}&space;=&space;S_a&space;&plus;&space;S_b&space;&plus;&space;9&space;k,&space;k&space;\in&space;\mathbb{Z}"/>.
+<img src="https://latex.codecogs.com/svg.latex?\forall&space;a,&space;b&space;\in&space;\mathbb{N},&space;S_{a&space;&plus;&space;b}&space;\equiv&space;S_a&space;&plus;&space;S_b&space;\;$$\textup{mod}\;&space;9&space;$$"/>
+
+This conjecture can be generalized for any _positional number system_. Check out proof.pdf for more information.
+
+## The Problem - Programmers Verison
+
+Let S(n: uint) -> uint be the sum of the digits of n in [_base 10_](https://en.wikipedia.org/wiki/Decimal). Then, for all a and b of type uint, S(a + b) - ( S(a) + S(b) ) % 9 == 0.

## Performance
-The conjecture was [proved by exhaustion](https://en.wikipedia.org/wiki/Proof_by_exhaustion) for the interval <img src="https://latex.codecogs.com/gif.latex?[0;10^4]"/> in multiple language implementations. The performance of each language was then avaliated as the following:
+The conjecture was [proved by exhaustion](https://en.wikipedia.org/wiki/Proof_by_exhaustion) for the interval <img src="https://latex.codecogs.com/svg.latex?[0,&space;10^4]"/> in multiple language implementations. The performance of each language was then avaliated as the following:

|Language      |Milliseconds|
|--------------|------------|
-|**Rust**      |115         |
+|**Rust**      |103         |
|**Kotlin**    |137         |
|**Go**        |155         |
|**C**         |271         |