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Conjecture: We have a(n) = 0 if p_n == 3 (mod 4).

For an odd prime p and integers c and d, Zhi-Wei Sun defined (c,d)_p to be the determinant of the (p-1) X (p-1) matrix whose (i,j)-entry is the Legendre symbol ((i^2+c*i*j+d*j^2)/p). It is easy to see that (-c,d)_p = (-1/p)*(c,d)_p. Sun conjectured that for any integer c and nonzero integer d there are infinitely many odd primes p with (c,d)_p = 0, moreover (c,d)_p = 0 if (d/p) = -1. He also formulated some concrete conjectures in the case (d/p) = 1. For example, (3,3)_p = 0 if p == 11 (mod 12), and (10,9)_p = 0 if p == 5 (mod 12); (3,2)_p = (4,2)_p = 0 if p == 7 (mod 8).

On August 12 2013, Zhi-Wei Sun conjectured that for any odd prime p and integers c and d with d not divisible by p, if (c,d)_p is nonzero then its p-adic valuation (i.e., p-adic order) must be even.