Block designs are called tight if their sizes achieve the Fisher type lower bound ${v \choose s}$. Symmetric $2$-designs, the Witt $4$-$(23, 7, 1)$ design and the Witt $4$-$(23, 16, 52)$ design are tight designs. It has been widely conjectured since 1970s that there are no other nontrivial tight designs. In this talk, we will give a proof of this conjecture. Estimates on prime gaps are used. If time permits, we will also discuss some applications of the technique to some codes.