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Conjecture: If p_n == 1 (mod 4), then a(n) == ((p_n-1)/2)! (mod p_n). If p_n == 3 (mod 4), then a(n) == (2/p_n) (mod p_n).

Zhi-Wei Sun also made the following general conjecture:

Let p be any odd prime. For each integer d let R(d,p) be the determinant of the (p+1)/2-by-(p+1)/2 matrix whose (i,j)-entry (i,j = 0,...,(p-1)/2) is the Legendre symbol ((i+d*j)/p). When p == 3 (mod 4), we have R(d,p) == (2/p) (mod p) if (d/p) = 1, and R(d,p) == 1 (mod p) if (d/p) = -1. In the case p == 1 (mod 4), we have R(c^2*d,p) == (c/p)*R(d,p) (mod p) for any integer c, and R(d,p) == 1 or -1 (mod p) if (d/p) = -1.

The author could prove that for any odd prime p and integer d not divisible by p, the determinant of the (p-1)-by-(p-1) matrix with (i,j)-entry (i,j=1,...,p-1) being the Legendre symbol ((i+dj)/p) has the exact value (-d/p)*p^{(p-3)/2}.

On August 19 2013, Zhi-Wei Sun found a formula for a(n). Namely, he made the following conjecture: If p_n == 1 (mod 4) and e(p_n)^{h(p_n)} = (a_n + b_n*sqrt(p_n))/2 with a_n and b_n integers of the same parity (where e(p_n) and h(p_n) are the fundamental unit and the class number of the quadratic field Q(sqrt(p_n)) respectively), then a(n) = - (2/p_n)*2^{(p_n-3)/2}*a_n. If p_n > 3 and p_n == 3 (mod 4), then a(n) = 2^{(p_n-1)/2}.

On August 19 2013, Zhi-Wei Sun proved all the conjectured congruences mentioned above by using the identity D(c,d,n) = (-d)^{n*(n+1)/2}*(n!)^{n+1}, where D(c,d,n) is the (n+1) X (n+1) determinant with (i,j)-entry equal to (i+d*j+c)^n for all i,j = 0,...,n. For any prime p == 1 (mod 4) he showed that R(d,p) == (d*(d/p))^{(p-1)/4}*((p-1)/2)! (mod p). Note also that the formula for a(n) found by Sun on August 9, 2013 is actually equivalent to Chapman's result on the evaluation of the determinant |((i+j-1)/p)|_{i,j=1,...,(p+1)/2}.

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.

Conjecture: a(n) = 0 if p_n == 5 (mod 6).

For an odd prime p and integers c and d, Zhi-Wei Sun defined [c,d]_p to be the determinant of the p X p matrix with (i,j)-entry equal to 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. He also conjectured that if (c,d)_p (defined in the comments in A225611) is nonzero with d not divisible by p, then [c,d]_p/(c,d)_p equals (1-p)/(p-2) or (p-1)/2 according as p divides c^2-4*d or not. He also made some concrete conjectures, for example, [3,2]_p = 0 if p == 3 (mod 4); and [4,2]_p = 0 if p == 5 or 7 (mod 8).

On Aug 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.