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A225611
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Determinant of the (p_n-1) X (p_n-1) matrix with (i,j)-entry equal to the Legendre symbol ((i^2+6*i*j+j^2)/p_n), where p_n is the n-th prime.
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2
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0, -16, 0, 0, 0, 18939904, 0, 0, -600706205614080, 0, -3126394312091238400, 915844279166632469526048000, 0, 0, 1513783909437524991467008819200000000, 0, -6597762875255062617688526826958066024448000, 0, 0
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OFFSET
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2,2
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COMMENTS
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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.
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LINKS
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MATHEMATICA
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a[n_]:=Det[Table[JacobiSymbol[i^2+6*i*j+j^2, Prime[n]], {i, 1, (Prime[n]-1)}, {j, 1, (Prime[n]-1)}]]
Table[a[n], {n, 2, 20}]
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PROG
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(PARI) a(n) = my(p=prime(n)); matdet(matrix(p-1, p-1, i, j, kronecker(i^2+6*i*j+j^2, 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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