An integer n is p-balanced for p a prime if, among all nonzero binomial coefficients (n k) for k = 0, ..., n (mod p), there are equal numbers of quadratic residues and nonresidues (mod p). Let T_p be the set of integers n, 0<=n<=p - 1, that are p-balanced. Among all the primes <1000000, only those with p = 2, 3, and 11 have T_p = ∅. The following table gives the p-balanced integers for small primes p (OEIS A093755).

binomial coefficient