Khong ton tai $a,b$ thoa man tinh chat nhu vay. Vi neu c la uoc nguyen to cua $a$ thi tu $a^b=b^a$, dan toi $c$ cung la uoc cua $b$ va nguoc lai. Do vay, ta co the dat dat $$a=p_1^{n_1}\cdots p_k^{n_k}\\ b=p_1^{m_1}\cdots p_k^{m_k}$$ trong do $p_k$ la cac so nguyen to khac nhau. Vi $a^b=b^a$ nen $p_i^{b}=p_i^a$, nhu vay $a=b$ ( mau thuan).
Ta co $a^b=b^a$ suy ra $blna=alnb\Rightarrow \frac{lna}{a}=\frac{lnb}{b}$. Xet ham so $f(t)=\frac{lnt}{t}$ co $f'(t)=\frac{1-lnt}{t^2}$. Ta thay $f'(t)<0$ voi $t>2$ va $f'(t)>0$ voi $t\leq2$. Neu co $a\ne b, a,b\in \mathbb{Z}$ ma $f(a)=f(b)$ thi mot trong hai so phai nho hon hoac bang 2, so con lai lon hon 2. Thu cho $a=1,2$ dan toi ko co b thoa man. Vay vo nghiem