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A227971
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Determinant of the (p_n+1)/2 X (p_n+1)/2 matrix with (i,j)-entry (i,j=0,...,(p_n-1)/2) being the Legendre symbol((i+j)/p_n), where p_n is the n-th prime.
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8
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-1, 2, 8, 32, 96, -1024, 512, 2048, 40960, 32768, 1572864, -33554432, 2097152, 8388608, 234881024, 536870912, 20937965568, 8589934592, 34359738368, -73392401154048, 549755813888, 2199023255552, -8796093022208000, -1577385769486516224, 11258999068426240
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OFFSET
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2,2
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COMMENTS
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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}.
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LINKS
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EXAMPLE
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a(2) = -1 since the determinant |((i+j)/3)|_{i=0,1; j=0,1}| equals -1.
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MATHEMATICA
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a[n_] := Det[Table[JacobiSymbol[i+j, Prime[n]], {i, 0, (Prime[n]-1)/2}, {j, 0, (Prime[n]-1)/2}]]; Table[a[n], {n, 2, 30}]
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PROG
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(PARI) a(n) = my(p=prime(n)); matdet(matrix((p+1)/2, (p+1)/2, i, j, i--; j--; kronecker(i+j, p))); \\ Michel Marcus, Aug 25 2021
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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