高联考后诸葛亮

通过必要时重排 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}(a_i)$，设 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}x_1>x_2>\cdots>x_n$．则有

$$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}|a_1|=\left|\sum_{i=2}^na_i\frac{x_i!}{x_1!}\right|\le\sum_{i=2}^n\left|a_i\frac{x_i!}{x_1!}\right|\le\frac{1}{x_1}\sum_{i=2}^n|a_i|.$$

但在 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}x_1>\sum_{i=2}^n|a_i/a_1|$ 时不能成立，因此 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\max(x_1, \cdots, x_n)$ 有上界．

首先考虑 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}n$ 为奇数的情况．此时 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}2\mid(2^n-2)$，而且 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}(2^n-2)/n>2$，因此得证．

再考虑 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}n$ 为偶数的情况，此时有 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\frac{n}{2}\mid(2^{n-1}-1)$．若 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}n-1$ 为合数，令 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}n=pq+1$，且 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}p>q$，则 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}(2^p-1)\mid(2^{pq}-1)$，且 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}2^p-1>\frac n2$，得证．

最后考虑 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}n$ 为质数加 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}1$ 的情况．设 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}n=p+1=2q$，则 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}2^p\equiv1\pmod{q}$．设 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}2$ 在 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\mathbb{Z}_{q}^\times$ 中的阶为 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}d$，则 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}d\mid p$ 且 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}d\le\vphi(q)\le q$，只可能 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}d=1$，但显然 $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}2^1\equiv2\not\equiv1\pmod{q}$. 矛盾，因此这种情况不存在．

综上，原命题得证．