jmc

Firstly, note that $29\equiv 1$ modulo 7, so $29^{13}\equiv 1$ modulo 7. Also, $5\equiv (-2)$, so $1-5^{13}\equiv 1+2^{13}$ modulo 7. Finally, $2^3\equiv 1$ modulo 7, so $2^{13}\equiv 2(2^3{)}^4\equiv 2\cdot 1\equiv 2$. Thus $29^{13}-5^{13}\equiv 1+2\equiv \boxed{3}$ modulo 7.