Periodic Occurance of Complete Intersection Monomial Curves

We study the complete intersection property of monomial curves in the family $\Gamma_{\aa + \jj} = {(t^{a_0 + j}, t^{a_1+j},…, t^{a_n + j}) ~ | ~ j \geq 0, ~ a_0 < a_1 <…< a_n}$\mathrm{.}Weprovethatif$\Gamma_{\aa+\jj}$ is a complete intersection for $j\gg 0$, then ${\mathrm{\Gamma }}_{\mathring{a}+\text{jj}+\underset{‾}{a_n}}$ is a complete intersection for $j\gg 0$. This proves a conjecture of Herzog and Srinivasan on eventual periodicity of Betti numbers of semigroup rings under translations for complete intersections. We also show that if ${\mathrm{\Gamma }}_{\mathring{a}+\text{jj}}$ is a complete intersection for $j\gg 0$, then ${\mathrm{\Gamma }}_{\mathring{a}}$ is a complete intersection. We also characterize the complete intersection property of this family when $n=3$.