It is known that $$F_2\equiv 2^{3r}\pmod{2^{3r+3}}\quad (r\ge 2),\quad F_3\equiv 3^{3r+1}\pmod{3^{3r+3}} \quad( r\ge 1),$$ $$F_p\equiv \Delta_p p^{3r+2}\pmod{p^{3r+3}} \quad (p\ge 5, r\ge 2),$$ where $\Delta_p$ depends only on $p$ (see Jacobsthal, Zahlentheoretische Eigenschaften der Binomialkoeffizienten, Norske Vid. Selsk., Skr. 1942, No. 4, 28 S. (1945).). More generally $$\binom{np}{kp}\equiv\binom{n}{k}\pmod{p^{3+\mathrm{ord}_pn+\mathrm{ord}_pk+\mathrm{ord}_p(n-k)+\mathrm{ord}_p\binom{n}{k}}}.$$ It is also known, see Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers that the last exponent can only be increased if $p$ devides $B_{p-3}$, the $(p-3)$rd Bernoulli number.

It is known that $$F_2\equiv 2^{3r}\pmod{2^{3r+3}}\quad (r\ge 2),\quad F_3\equiv 3^{3r+1}\pmod{3^{3r+3}} \quad( r\ge 1),$$ $$F_p\equiv \Delta_p p^{3r+2}\pmod{p^{3r+3}} \quad (p\ge 5, r\ge 2),$$ where $\Delta_p$ depends only on $p$ (see Jacobsthal, Zahlentheoretische Eigenschaften der Binomialkoeffizienten, Norske Vid. Selsk., Skr. 1942, No. 4, 28 S. (1945).). More generally $$\binom{np}{kp}\equiv\binom{n}{k}\pmod{p^{3+\mathrm{ord}_pn+\mathrm{ord}_pk+\mathrm{ord}_p(n-k)+\mathrm{ord}_p\binom{n}{k}}}.$$ It is also known, see Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers that the last exponent can only be increased if $p$ devides $B_{p-3}$, the $(p-3)$rd Bernoulli number.

See also discussion at Binomial supercongruences: is there any reason for them?

It is known that $$F_2\equiv 2^{3r}\pmod{2^{3r+3}}\quad (r\ge 2),\quad F_3\equiv 3^{3r+1}\pmod{3^{3r+3}} \quad( r\ge 1),$$ $$F_p\equiv \Delta_p p^{3r+2}\pmod{p^{3r+3}} \quad (p\ge 5, r\ge 2),$$ where $\Delta_p$ depends only on $p$ (see Jacobsthal, Zahlentheoretische Eigenschaften der Binomialkoeffizienten, Norske Vid. Selsk., Skr. 1942, No. 4, 28 S. (1945).). More generally $$\binom{np}{kp}\equiv\binom{n}{k}\pmod{p^{3+\mathrm{ord}_pn+\mathrm{ord}_pk+\mathrm{ord}_p(n-k)+\mathrm{ord}_p\binom{n}{k}}}.$$

It is known that $$F_2\equiv 2^{3r}\pmod{2^{3r+3}}\quad (r\ge 2),\quad F_3\equiv 3^{3r+1}\pmod{3^{3r+3}} \quad( r\ge 1),$$ $$F_p\equiv \Delta_p p^{3r+2}\pmod{p^{3r+3}} \quad (p\ge 5, r\ge 2),$$ where $\Delta_p$ depends only on $p$ (see Jacobsthal, Zahlentheoretische Eigenschaften der Binomialkoeffizienten, Norske Vid. Selsk., Skr. 1942, No. 4, 28 S. (1945).). More generally $$\binom{np}{kp}\equiv\binom{n}{k}\pmod{p^{3+\mathrm{ord}_pn+\mathrm{ord}_pk+\mathrm{ord}_p(n-k)+\mathrm{ord}_p\binom{n}{k}}}.$$ It is also known, see Granville, Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers that the last exponent can only be increased if $p$ devides $B_{p-3}$, the $(p-3)$rd Bernoulli number.