jmc

$$9997^2=(10^4-3{)}^2=10^8-2\cdot 3\cdot 10^4+9.$$We can factor out $10^4$ from the first two terms to make the computation easier: $$9997^2=10^4(10^4-6)+9=10^4\cdot 9994+9=\boxed{99940009}.$$