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Garrett Ervin's answer to When is $A$ isomorphic to $A^3$? mentions also some results on topological spaces. (Although the question was originally about abelian groups.)

The results mentioned there seem to answer your question - although I do hope that somebody can provide a more elementary solution. (The result from these papers have several additional requirements on the space $X$.)

In particular, the linked answer mentions the paper A. Orsatti and N. Rodino, Homeomorphisms between finite powers of topological spaces, Topology Appl. 23 (1986), no. 3, 271--277; MR858335, Zbl 0603.54009.

Let $\lambda$ be an infinite cardinal number. It is proved that, for each positive integer $r$, there exists a compact connected homogeneous topological space $X$ of weight $\lambda$ such that $X^n$ is homeomorphic to $X^m$ iff $n\equiv m \pmod r$. The cardinality of the set of homeomorphism classes of compact connected homogeneous spaces with this property is exactly $2^\lambda$. Moreover every completely regular space of weight $\lambda$ is embeddable in a space of this type.

Another paper with related results of this type is Věra Trnková: Products of metric, uniform and topological spaces. Commentationes Mathematicae Universitatis Carolinae, vol. 31 (1990), issue 1, pp. 167-180; MR1056184, Zbl 0696.54009.

For every triple of natural number $a$, $b$, $c$ there exists a metric space $X$, the $m$-th power and the $n$-th power of which are

• homeomorphic iff $m\equiv n \pmod a$
• uniformly homeomorphic iff $m\equiv n \pmod {ab}$
• isomoteric iff $m\equiv n \pmod {abc}$

This is a consequence of the main theorem proved in the present paper, where simultaneous representations of commutative semigroups by the products of metric, uniform and topological spaces are investigated.

Garrett Ervin's answer to When is $A$ isomorphic to $A^3$? mentions also some results on topological spaces. (Although the question was originally about abelian groups.)

The results mentioned there seem to answer your question - although I do hope that somebody can provide a more elementary solution. (The result from these papers have several additional requirements on the space $X$.)

In particular, the linked answer mentions the paper A. Orsatti and N. Rodino, Homeomorphisms between finite powers of topological spaces, Topology Appl. 23 (1986), no. 3, 271--277; MR858335, Zbl 0603.54009.

Let $\lambda$ be an infinite cardinal number. It is proved that, for each positive integer $r$, there exists a compact connected homogeneous topological space $X$ of weight $\lambda$ such that $X^n$ is homeomorphic to $X^m$ iff $n\equiv m \pmod r$. The cardinality of the set of homeomorphism classes of compact connected homogeneous spaces with this property is exactly $2^\lambda$. Moreover every completely regular space of weight $\lambda$ is embeddable in a space of this type.

Another paper with related results of this type is Věra Trnková: Products of metric, uniform and topological spaces. Commentationes Mathematicae Universitatis Carolinae, vol. 31 (1990), issue 1, pp. 167-180; MR1056184, Zbl 0696.54009.

For every triple of natural number $a$, $b$, $c$ there exists a metric space $X$, the $m$-th power and the $n$-th power of which are

• homeomorphic iff $m\equiv n \pmod a$
• uniformly homeomorphic iff $m\equiv n \pmod {ab}$
• isometric iff $m\equiv n \pmod {abc}$

This is a consequence of the main theorem proved in the present paper, where simultaneous representations of commutative semigroups by the products of metric, uniform and topological spaces are investigated.

Garrett Ervin's answer to When is $A$ isomorphic to $A^3$? mentions also some results on topological spaces. (Although the question is about topological groups.)

The results mentioned there seem to answer your question - although I do hope that somebody can provide a more elementary solution. (The result from these papers have several additional requirements on the space $X$.)

In particular, the linked answer mentions the paper A. Orsatti and N. Rodino, Homeomorphisms between finite powers of topological spaces, Topology Appl. 23 (1986), no. 3, 271--277; MR858335, Zbl 0603.54009.

Let $\lambda$ be an infinite cardinal number. It is proved that, for each positive integer $r$, there exists a compact connected homogeneous topological space $X$ of weight $\lambda$ such that $X^n$ is homeomorphic to $X^m$ iff $n\equiv m \pmod r$. The cardinality of the set of homeomorphism classes of compact connected homogeneous spaces with this property is exactly $2^\lambda$. Moreover every completely regular space of weight $\lambda$ is embeddable in a space of this type.

Another paper with related results of this type is Věra Trnková: Products of metric, uniform and topological spaces. Commentationes Mathematicae Universitatis Carolinae, vol. 31 (1990), issue 1, pp. 167-180; MR1056184, Zbl 0696.54009.

For every triple of natural number $a$, $b$, $c$ there exists a metric space $X$, the $m$-th power and the $n$-th power of which are

• homeomorphic iff $m\equiv n \pmod a$
• uniformly homeomorphic iff $m\equiv n \pmod {ab}$
• isomoteric iff $m\equiv n \pmod {abc}$

This is a consequence of the main theorem proved in the present paper, where simultaneous representations of commutative semigroups by the products of metric, uniform and topological spaces are investigated.

Garrett Ervin's answer to When is $A$ isomorphic to $A^3$? mentions also some results on topological spaces. (Although the question was originally about abelian groups.)

The results mentioned there seem to answer your question - although I do hope that somebody can provide a more elementary solution. (The result from these papers have several additional requirements on the space $X$.)

In particular, the linked answer mentions the paper A. Orsatti and N. Rodino, Homeomorphisms between finite powers of topological spaces, Topology Appl. 23 (1986), no. 3, 271--277; MR858335, Zbl 0603.54009.

Let $\lambda$ be an infinite cardinal number. It is proved that, for each positive integer $r$, there exists a compact connected homogeneous topological space $X$ of weight $\lambda$ such that $X^n$ is homeomorphic to $X^m$ iff $n\equiv m \pmod r$. The cardinality of the set of homeomorphism classes of compact connected homogeneous spaces with this property is exactly $2^\lambda$. Moreover every completely regular space of weight $\lambda$ is embeddable in a space of this type.

Another paper with related results of this type is Věra Trnková: Products of metric, uniform and topological spaces. Commentationes Mathematicae Universitatis Carolinae, vol. 31 (1990), issue 1, pp. 167-180; MR1056184, Zbl 0696.54009.

For every triple of natural number $a$, $b$, $c$ there exists a metric space $X$, the $m$-th power and the $n$-th power of which are

• homeomorphic iff $m\equiv n \pmod a$
• uniformly homeomorphic iff $m\equiv n \pmod {ab}$
• isomoteric iff $m\equiv n \pmod {abc}$

This is a consequence of the main theorem proved in the present paper, where simultaneous representations of commutative semigroups by the products of metric, uniform and topological spaces are investigated.

Garrett Ervin's answer to When is $A$ isomorphic to $A^3$? mentions also some results on topological spaces. (Although the question is about topological groups.)

The results mentioned there seem to answer your question - although I do hope that somebody can provide a more elementary solution. (The result from these papers have several additional requirements on the space $X$.)

In particular, the linked answer mentions the paper A. Orsatti and N. Rodino, Homeomorphisms between finite powers of topological spaces, Topology Appl. 23 (1986), no. 3, 271--277; MR858335, Zbl 0603.54009.

Let $\lambda$ be an infinite cardinal number. It is proved that, for each positive integer $r$, there exists a compact connected homogeneous topological space $X$ of weight $\lambda$ such that $X^n$ is homeomorphic to $X^m$ iff $n\equiv m \pmod r$. The cardinality of the set of homeomorphism classes of compact connected homogeneous spaces with this property is exactly $2^\lambda$. Moreover every completely regular space of weight $\lambda$ is embeddable in a space of this type.

Another paper with related results of this type is Vìra Trnková: Products of metric, uniform and topological spaces. Commentationes Mathematicae Universitatis Carolinae, vol. 31 (1990), issue 1, pp. 167-180; MR1056184, Zbl 0696.54009.

For every triple of natural number $a$, $b$, $c$ there exists a metric space $X$, the $m$-th power and the $n$-th power of which are

• homeomorphic iff $m\equiv n \pmod a$
• uniformly homeomorphic iff $m\equiv n \pmod {ab}$
• isomoteric iff $m\equiv n \pmod {abc}$

This is a consequence of the main theorem proved in the present paper, where simultaneous representations of commutative semigroups by the products of metric, uniform and topological spaces are investigated.

Garrett Ervin's answer to When is $A$ isomorphic to $A^3$? mentions also some results on topological spaces. (Although the question is about topological groups.)

The results mentioned there seem to answer your question - although I do hope that somebody can provide a more elementary solution. (The result from these papers have several additional requirements on the space $X$.)

In particular, the linked answer mentions the paper A. Orsatti and N. Rodino, Homeomorphisms between finite powers of topological spaces, Topology Appl. 23 (1986), no. 3, 271--277; MR858335, Zbl 0603.54009.

Let $\lambda$ be an infinite cardinal number. It is proved that, for each positive integer $r$, there exists a compact connected homogeneous topological space $X$ of weight $\lambda$ such that $X^n$ is homeomorphic to $X^m$ iff $n\equiv m \pmod r$. The cardinality of the set of homeomorphism classes of compact connected homogeneous spaces with this property is exactly $2^\lambda$. Moreover every completely regular space of weight $\lambda$ is embeddable in a space of this type.

Another paper with related results of this type is Věra Trnková: Products of metric, uniform and topological spaces. Commentationes Mathematicae Universitatis Carolinae, vol. 31 (1990), issue 1, pp. 167-180; MR1056184, Zbl 0696.54009.

For every triple of natural number $a$, $b$, $c$ there exists a metric space $X$, the $m$-th power and the $n$-th power of which are

• homeomorphic iff $m\equiv n \pmod a$
• uniformly homeomorphic iff $m\equiv n \pmod {ab}$
• isomoteric iff $m\equiv n \pmod {abc}$

This is a consequence of the main theorem proved in the present paper, where simultaneous representations of commutative semigroups by the products of metric, uniform and topological spaces are investigated.