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Abstract Let p be a prime with p≡1mod8 $p\equiv1\bmod8$, ψ be an eighth character modp, and τ(ψ) $\tau(\psi)$ denote the classical Gauss sum modp. The main purpose of this paper is using the analytic method and the properties of the classical Gauss sum to study the computational problem of one kind rational polynomial of τ(ψ) $\tau(\psi)$. In the end, we prove an interesting second-order linear recursive formula for it.