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.

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.

These days I'm trying to research what's about of 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 literarture, 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 to 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 to 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 arangement of $n=A\cdot B$ unit squares. I emphasize with this 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 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've the following obvious statement.

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

Thus we must to deal with squares $S=N\times N$ with $N>7$ an odd integer. Exploting a specialization of a well-known identity $1+3+5=3^2$ I can to tessellate the square $9\times 9$, because this square can be splitted 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 tesellated 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 to get it, and it is required, I can to add in a figure some of the mentioned tessellations.

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.