Noether symmetries in f(T, T _G) cosmology

All degrees of freedom related to the torsion scalar can be explored by analysing, $f(T,T_G)$ gravity formalism where, $T$ is a and $T_G$ teleparallel counterpart Gauss-Bonnet topological invariant term. The well-known Noether symmetry approach useful tool for selecting models that are motivated at fundamental level determining exact solution given Lagrangian, hence we explore in with three different forms study how establish nontrivial vector form each one them. We extend analysis made \cite{capozziello2016noether} $f(T,T_{G})=b_{0}T_{G}^{k}+t_{0}T^{m}$ discussed this model linear equivalent term, followed two containing exponential have shown all cases will allow us obtain non-trivial which play an important role solutions cosmological equations.