Problem 478

Erdős Problem #478: declared status 'open'. Formalized: no. Let $p$ be a prime and$$A_p = \{ k! \pmod{p} : 1\leq k<p\}.$$Is it true that$$\lvert A_p\rvert \sim (1-\tfrac{1}{e})p?$$ Current best: The best known lower bound is due to Grebennikov, Sagdeev, Semchankau, and Vasilevskii [GSSV24],$$\lvert A_p\rvert \geq (\sqrt{2}-o(1))p^{1/2},$$which follows from proving that $\lvert A_pA_p\rvert=(1+o(1))p$. Wilson's theorem implies $(p-2)!\equiv 1\pmod{p}$, and hence $\lvert A_p\rvert\leq p-2$. Prize: no. OEIS: A210184. Tags: factorials, number theory.