A complete classification of permutation binomials of the form $X^r(X^{q-1}+a)$ over finite fields

We classify, for every prime power $q$ and every $e⩾2$, the permutation binomials $X^r(X^{q-1}+a)$ over ${\mathbb{F}}_{q^e}$. Writing ${\mathrm{\ell }}_j(q)=(q^j-1)\mathrm{/}(q-1)$, such a binomial is a permutation if and only if $\gcd (r,q-1)=1$, $(-a{)}^{{\mathrm{\ell }}_e(q)}\mathrm{\ne }1$, and $r{\mathrm{\ell }}_h(q)\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}{\mathrm{\ell }}_e(q))$ for some $1⩽h<e$ coprime to $e$. This proves a conjecture of Masuda, Rubio, and Santiago: every permutation binomial of this form arises from $(X^{q^h}+aX)\circ X^r$ for a suitable $h$. We also determine the exact number of distinct permutation functions represented by this family. As a further consequence, we completely classify the broader family $X^r(X^{d(q-1)}+a)$ in the coprime-index case $\gcd (d,{\mathrm{\ell }}_e(q))=1$. The new ingredient in the main classification is the necessity argument: selected Hermite power sums are organized so that Lucas’ theorem turns their coefficients into digit conditions; a Farey-guided local argument then forces successive base-$q$ digits, and cyclic rotations yield the inverse congruence. In characteristic $2$, a mod-$4$ lift to an auxiliary ring retains endpoint information lost modulo $2$.