Are the rings $k[[t^3,t^4,t^5]]$ and $k[[t^4,t^5,t^6]]$ Gorenstein? (Matsumura, Exercise 18.8)

Let $k$ be a field and $R$ a graded $k$-algebra. Then $R$ is Gorenstein iff $R_m$ is Gorenstein, where $m$ is the irrelevant maximal ideal of $R$. (This is exercise 3.6.20(c) from Bruns & Herzog.)

If $R$ is a Noetherian local ring, then $R$ is Gorenstein iff its completion $\widehat{R}$ is Gorenstein. (This is Proposition 3.1.19(c) from Bruns & Herzog.)

Let $k$ be a field and $S$ a numerical semigroup. Then $k[S]$ is Gorenstein iff $S$ is symmetric. (This is Theorem 4.4.8 from Bruns & Herzog.)

The examples from Matsumura are completions of affine semigroup rings with respect to their irrelevant maximal ideals. For instance, $k[[t^3,t^4,t^5]]$ is Gorenstein iff $k[t^3,t^4,t^5]$ is Gorenstein iff $S=\langle 3,4,5\rangle$ is symmetric and this is not the case. On the other side, in the second example $S=\langle 4,5,6\rangle$ is symmetric.