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I think about unary languages $L_k$, where $L_k$ is set of all words which length is the sum of $k$ squares. Formally: $$L_k=\{a^n\mid n=\sum_{i=1}^k {n_i}^2,\;\;n_i\in\mathbb{N_0}\;(1\le i\le k)\} $$ It is easy to show that $L_1=\{a^{n^2}\mid n\in\mathbb{N_0}\}$ is not regular (e.g. with Pumping-Lemma).
Further, we know that each natural number is the sum of four squares which implies that for $k\ge 4$ all languages $L_k$ are regular since $L_k=L(a^*)$.

Now, I am interested in the cases $k=2$ and $k=3$:

$L_2=\{a^{{n_1}^2+{n_2}^2}\mid n_1,n_2\in\mathbb{N_0}\}$, $L_3=\{a^{{n_1}^2+{n_2}^2+{n_3}^2}\mid n_1,n_2,n_3\in\mathbb{N_0}\}$.

Unfortunately, I am not able to show whether this languages are regular or not (even with the help of Legendre's three-square theorem or Fermat's theorem on sums of two squares).

I am pretty sure that at least $L_2$ is not regular but unhappily thinking is not a proof. Any help?

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Let's start with $L_2$. It is known that the upper density of integers which are the sum of two squares is 0. If $L_2$ were regular then it would be eventually periodic, and so, since its upper density is 0, finite. But we know that there are arbitrarily large integers in $L_2$, so that $L_2$ cannot be regular.

Regarding $L_3$, consider the words $w_k = 1^{4^k 7}$. I claim that for $k < \ell$, the words $w_k,w_\ell$ are inequivalent. Indeed, $w_k 1^{4^k 8} \notin L_3$ while $w_\ell 1^{4^k 7} \in L_3$. The Myhill–Nerode criterion then shows that $L_3$ is irregular.

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Suppose $L_3$ is regular. Then so is its complement, which by Legendre's three-square theorem is $\{a^n\ |\ n=4^k(8l+7), k,l\in\mathbb{N}\}$. By Parikh's theorem, this would imply that the set of lengths $S=\{4^k(8l+7)\ |\ k,l\in\mathbb{N}\}$ is semi-linear, i.e. a finite union $\bigcup_{i=1}^NS_i$ of linear sets $S_i=\{a_i + rb_i\ |\ r\in\mathbb{N}\}$.

Consider two elements $s_1 = 4^{k_1}(8l_1+7), s_2 = 4^{k_2}(8l_2+7)\in S$ with $k_1>k_2$, and let $r:=k_1-k_2$. If $s_1,s_2$ are both in the same $S_i$, then so is either $2s_1-s_2$ or $2s_2-s_1$ (depending on whether $s_1<s_2$ or $s_1>s_2$). But

  • $2(4^{k_1}(8l_1+7))-(4^{k_2}(8l_2+7)) = 4^{k_2}(8l'-7)$, where $l'=4^{r-1}(8l_1+7)-l_2$,
  • $2(4^{k_2}(8l_2+7))-(4^{k_1}(8l_1+7)) = 4^{k_2}(8l'-7*4^r+14)$, where $l'=2l_2-4^rl_1$.

Neither of these are in $S$, so $s_1,s_2$ would have to be in different members of the union. But this is impossible, since $S$ is a finite union, and there are infinitely many different $k$.

Therefore, $L_3$ is not regular.

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