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Conjecture: p_n never divides a(n), and moreover -a(n) is a quadratic residue mod p_n.

Zhi-Wei Sun also made the following conjecture:

Let p be any odd prime. For each integer d let S(d,p) be the determinant of the (p-1)/2 X (p-1)/2 matrix whose (i,j)-entry is the Legendre symbol ((i^2+d*j^2)/p). If d is a quadratic residue mod p, then so is -S(d,p). If d is a quadratic non-residue mod p, then we have S(d,p) = 0.

These were proved in version 9 of arXiv:1308.2900 (2018). In addition, the author has the following new conjecture.

Conjecture: For any prime p == 3 (mod 4), the number -S(1,p) is a positive square divisible by 2^((p-3)/2), i.e., -S(1,p) = (2^((p-3)/4)*m)^2 for some positive integer m. - Zhi-Wei Sun, Sep 09 2018

Conjecture: 2*a(n) is always a quadratic residue mod p_n.

Zhi-Wei Sun also made the following general conjecture:

Let p be any odd prime. For each integer d let T(d,p) be the determinant of the (p+1)/2 X (p+1)/2 matrix whose (i,j)-entry (i,j=0,...,(p-1)/2) is the Legendre symbol ((i^2+d*j^2)/p). Then T(-d,p) == (2/p)*T(d,p) (mod p). If d is a quadratic residue mod p, then so is 2*T(d,p). If d is a quadratic non-residue mod p, then T(d,p) is a quadratic residue mod p.

The author has proved the following results for any odd prime p:

(1) If c and d are integers with c not divisible by p, and d' == c^2*d (mod p), then T(d',p) = (c/p)^((p+1)/2)*T(d,p).

(2) If p is congruent to 1 mod 4 and d is not divisible by p, then T(-d,p) = (2/p)*T(d,p).

The general conjecture mentioned above was proved in a recent preprint of Zhi-Wei Sun.

Conjecture: a(n) = 0 if and only if p_n == 3 (mod 4).

Note that for an odd prime p we have (((p-1)/2)!)^2 == (-1)^{(p+1)/2} (mod p) by Wilson's theorem. In 1961, Mordell proved that((p-1)/2)! == (-1)^{(h(-p)+1)/2} (mod p) for any prime p > 3 with p == 3 (mod 4), where h(-p) is the class number of the imaginary quadratic field Q(sqrt(-p)).

Conjecture: We have (a(n)/p_n) = ((2(-1)^{(p_n-1)/2}-4)/p_n).

Zhi-Wei Sun also made the following general conjectures for any odd prime p:

(i) Let c and d be integers not divisible by p, and let S be the determinant of the (p-1)/2-by-(p-1)/2 matrix whose (i,j)-entry is the Legendre symbol ((i^2+d*j^2+c)/p).

Then (S/p) = 1 if (c/p) = 1 and (d/p) = -1; (S/p) = (-1/p) if (c/p) = (d/p) = -1; (S/p) = (-2/p) if (-c/p) = (d/p) = 1; and (S/p) = (-6/p) if (-c/p)= -1 and (d/p) = 1.

(ii) Let d be any integer not divisible by p. For k = 1,2 let D_k(c) be the determinant of the (p+1)/2-by-(p+1)/2 matrix with (i,j)-entry (i,j = 0,...,(p-1)/2) being the Legendre symbol ((i^k+d*j^k+c)/p). Then D_k(c) == D_k(0) (mod p) for every integer c.

Conjecture: In the case p_n == 1 (mod 4), (2/p_n)*a(n) is a positive odd integer whose prime factors are all congruent to 1 modulo 4, and moreover for some integer b(n) we have b(n) + (2/p_n)*a(n)*sqrt(p_n) = e(p_n)^{(2-(2/p_n))h(p_n)}, where e(p_n) and h(p_n) are the fundamental unit and the class number of the real quadratic field Q(sqrt(p_n)) respectively.

Note that a(n) = 0 when p_n == 3 (mod 4), this is because the transpose of the determinant a(n) coincides with (-1/p_n)^{(p_n-1)/2}*a(n) = -a(n).

M. Vsemirnov has proved Robin Chapman's conjecture on the evaluation of the determinant of the (p+1)/2-by-(p+1)/2 matrix with (i,j)-entry (i,j = 0,...,(p-1)/2) being the Legendre symbol ((j-i)/p), where p is an odd prime.

On Aug 14 2013, Robin Chapman informed the author that he first made the conjecture about the exact value of a(n) in a private manuscript dated Aug 05 2003.

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.