let $p$ be postive integer, if $$(2p+1)\mid\left(\binom{2p}{p}+(-1)^{p-1}\right)$$

I conjecture $2p+1$ is prime number.

I have test $p=1,2,3,5,6,8,9,11,14,15$ is hold,$2p+1=3,5,7,11,13,17,19,23,29,31$ are prime,and

$$\frac{\binom{2}{1}+1}{3}=1,\frac{\binom{4}{2}-1}{5}=1,\frac{(\binom{6}{3}+1)}{7}=3,\frac{(\binom{10}{5}+1)}{11}=23,\frac{\binom{12}{6}-1}{13}=71,\frac{\binom{16}{8}-1}{17}=757,\frac{\binom{18}{9}+1}{19}=2559,\frac{\binom{22}{11}+1}{23}=30671,\frac{\binom{28}{14}-1}{29}=1383331,\frac{\binom{30}{15}+1}{31}=5003791$$

if this true,then we have found prime closed-form formula?