These days I'm trying to research relations between prime numbers and the notion of chirality in the $xy$-plane. Wikipedia has the article *Chirality*.

I don't know if this relation or the problem for which I ask here is in the literature; please add a comment in such case.

**Definition.** In this post, I consider a definition ad hoc, defining as **admissible polygons** $P$ the connected regions in the $xy$-lattice that are arrangements of unit squares with $\text{area}(P)=p>3$ a prime number, and also by definition I consider that these polygons must have a chiral shape, that is these pieces are chirals. A tessellation of a connected region $R$ by means of admissible polygons is called an **admissible tessellation of** $R$ (in our problem $R=S$ will be a square $S$ in the $xy$-lattice).

(The rationale for previous definitions is that any polygonal representation of the integers $1,2$ or $3$ aren't chirals, and on the other hand, again by definition one can consider that a given composite integer can be represented as a rectangle or square $A\times B$, that isn't chiral polygon, of sides $1<A\leq B$ in the $xy$-lattice as an arrangement of $n=A\cdot B$ unit squares. I emphasize with these brackets that these pieces will not be used in the tessellations in this post).

**Conjecture.** *There exists a constant* $N_0$ *such that every square* $S=N\times N$ *of side* $N$ *with* $N>N_0$ *can be represented by admissible polygons.*

Question.I would like to know what work can be done to prove the veracity of the previous conjecture.Many thanks.

With admissible polygons it is easy get admissible tessellations for the square $4\times 4$, $5\times 5$, $6\times 6$ and $7\times 7$. We have the following obvious statement.

**Proposition.** *Given an admissible tessellation for the square* $S=N\times N$ *of* $N^2$ *unit squares, it is immediate to get an admissible tessellation for the square* $\hat{S}$ *of side* $2N$, *since this second square* $\hat{S}$ *admits the obvious decomposition into four squares* $S$.

Thus we must deal with squares $S=N\times N$ with $N>7$ an odd integer. Exploiting a specialization of a well-known identity $1+3+5=3^2$ I can tessellate the square $9\times 9$, because this square can be split as two copies of the square $(1+3)\times(1+3)$, plus the tessellated region corresponding to the square $5\times 5$ plus a region that can be tessellated with two admissible polygons with areas $7$ and $17$, being these chirals (I just emphasize it to provide details).

I think that the conjecture isn't obvious.

If you have problems getting it, and it is required, I can add in a figure some of the mentioned tessellations.

4426156on MSE), the post was deleted by the botCommunity17 days ago. An user provided a possible approach adding a comment in the deleted post. $\endgroup$andchirality. But I'm really interesting to continue what are the relations between primes and chirality.If it was confussing now, I'm agree because I'm trying to research possible relations:I apologize it (I'm sorry), many thanks. $\endgroup$Arithmetic billiards. Which I good like is to characterize chirality by means of some property of rays of light for certain rectangles (these rectangles representing prime numbers). This is an idea/speculation that I'm trying in my home (for me, prime numbers $p>2$ are similar than the capital letterL, that is the representation $1$ squared unit plus a rectangle of dimensions $1\times \varphi(p)$, where $\varphi(p)$ is the Euler totient function. But I don't know how motivate this idea/specualtion. Please feel free to explore it @StevenStadnicki $\endgroup$2more comments