= Solution
Choose longitudinal current along $x$, <magnetic field> along $z$, and define the oriented <Hall coefficient> by $R_H=E_y/(j_xB_z)$. For one carrier type with signed charge $q$, a stationary transverse force balance is $q(\mathbf E+\mathbf v\times\mathbf B)_y=0$. Since $(\mathbf v\times\mathbf B)_y=-v_xB_z$, this gives $E_y=v_xB_z$. Using $j_x=nqv_x$,
$$
\boxed{R_H=\frac1{nq}},\qquad R_H=-\frac1{ne}\ \text{for electrons},\quad R_H=+\frac1{pe}\ \text{for holes}.
$$
This is the weak-field isotropic one-carrier approximation with <Hall scattering factor> one, not a general multiband identity.
Under the stipulated electron-dominated approximation, the given Hall magnitudes imply
$$
n_i(300)=\frac1{e(625)}=9.99\times10^{15}\ \mathrm{m^{-3}},\qquad
n_i(500)=4\times10^4n_i(300)=3.99\times10^{20}\ \mathrm{m^{-3}}.
$$
Take the ratio of the intrinsic activation law:
$$
4\times10^4=\left(\frac{500}{300}\right)^{3/2}
\exp\left[\frac{E_g}{2k_B}\left(\frac1{300}-\frac1{500}\right)\right].
$$
Solving gives the model's activation-gap estimate
$$
\boxed{E_g=\frac{2k_B\ln[4\times10^4(300/500)^{3/2}]}{1/300-1/500}
=2.04\times10^{-19}\ \mathrm J=1.27\ \mathrm{eV}}.
$$
This result uses the supplied Hall-temperature ratio and the fixed-gap, fixed-prefactor assumptions. A temperature-dependent gap or changing Hall factors would modify its interpretation.
The source's description of <electrons> as intrinsic silicon's majority carriers needs qualification. \b[Intrinsic neutrality gives equal <electron> and hole concentrations], not an <electron density> majority. <Electrons> can nevertheless dominate conductivity and the Hall response because their mobility is larger. For two weak-field carriers,
$$
R_H=\frac{p\mu_h^2-n\mu_e^2}{e(p\mu_h+n\mu_e)^2}.
$$
At $n=p=n_i$ this becomes $R_H=-(\mu_e-\mu_h)/[en_i(\mu_e+\mu_h)]$. Thus $\mu_e>\mu_h$ explains the negative sign, while the one-electron formula is quantitatively accurate only if the hole contribution can be neglected. Equality $n=p$ itself is a concrete counterexample to a literal electron-number-majority claim. The numerical deductions above retain the approximation explicitly requested by the source.
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