Erratum to the paper: ''On the Diophantine equations $(x-1)^3+x^5+(x+1)^3=y^n$ and $(x-1)^5+x^3+(x+1)^5=y^n$''

In the proof of [2, Theorem 1.2], we miss the case $5|x$ for $n$ an odd prime. When $5\nmid x$, since $\gcd(2x^2+1, x)=1$, $\gcd(x^2+10, x)=1$ or $2$, one has $x=2^{\beta}u^n$, $19^\alpha z^n=2x^2+1=2^{2\beta+1}u^{2n}+1$, and then Lemma 2.3 can be used to get the result. But when $5|x$, since $\gcd(x^2+10,x)=5$ or $10$, one has $x=2^{\beta}\times5^{n-1}u^n$, $19^\alpha z^n=2x^2+1=2^{2\beta+1}\times5^{2n-2}u^{2n}+1$, and then the following lemma can be used to obtain the result. [...]