Consider an odd prime $p\equiv1 \pmod {16}$ and set $M=\frac{p-1}{2}$ for notational convenience. Then is there even a single prime $p$ of the above form for which the following congruence holds? $$\binom{M}{M/2}\binom{M}{M/4}\equiv \pm \binom{M}{3M/8}\binom{M}{7M/8}\pmod p ?$$
Here the symbol $\binom{n}{r}$ is a binomial coefficient and is defined as $\frac{n!}{r!(n-r)!}$
I am trying to prove something significantly more abstract but have managed to reduce it down to proving that the above statement can never hold. I have written some code to check this for primes of the above form less than $1000$ and found no counter examples. My attempts to find results on binomial coefficients $\pmod p$ have mostly involved Lucas' theorem which does not seem very applicable here. I would appreciate a solution, reference or possible strategies I might try.
from Hot Weekly Questions - Mathematics Stack Exchange
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