# Integers with a Hamiltonian Square Path

Let $$n>1$$ be an integer and set $$[n]=\{1,\ldots,n\}$$. We say that $$n$$ has a "Hamiltonian Square Path" if there is a bijection $$\varphi:[n]\to[n]$$ such that for all $$k\in [n-1]$$ we have that $$\varphi(k)+\varphi(k+1)$$ is a square number.

For instance $$15$$ and $$16$$ have this property.

Question. Is there an integer $$N>1$$ such that every integer $$n\geq N$$ has a Hamiltonian Square Path?

Note. This problem can be formulated in the language of graph theory and Hamiltonian paths. We say that $$a\neq b\in [n]$$ form an edge if their sum is square, and the above question is about integers such that the resulting graph has a Hamiltonian path.

Post #22 at https://mersenneforum.org/showthread.php?p=477787 by R. Gerbicz claims a proof that $$N=25$$ is the answer, and that for $$N\ge32$$ there is a Hamiltonian cycle. See also the tabulation and discussion at the Online Encyclopedia of Integer Sequences.