A177861 -id:A177861 - cp's OEIS frontend

A177860 Product of the quadratic residues of prime(n).

1, 1, 4, 8, 540, 12960, 1797120, 22619520, 465813504, 267346759680000, 216218419200000, 199658024013127680000, 136256285631578112000000, 12446179270879850496000000, 34611344543529418987929600

Offset: 1

Jonathan Sondow, May 14 2010

a(n) == (-1)^((p+1)/2) (mod p), if p = prime(n) is odd.

			The quadratic residues of prime(4) = 7 are 1, 2, and 4, so a(4) = 1*2*4 = 8.

Carl-Erik Froeberg, On sums and products of quadratic residues, BIT, Nord. Tidskr. Inf.-behandl. 11 (1971) 389-398.

Rahul Gupta, Algorithmic Number Theory, Section 24.5

Cf. A076409 Sum of the quadratic residues of prime(n), A177861 Product of the quadratic nonresidues of prime(n), A163366 Product of the quadratic residues of prime(n) modulo prime(n).

Table[ Apply[Times, Flatten[Position[ Table[JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], 1]]], {n, 1, 16}]

a(n) = (p-1)!/A177861(n), where p = prime(n).

A177863 Product modulo p of the quadratic nonresidues of p, where p = prime(n).

Original entry on oeis.org

1, 2, 1, 6, 10, 1, 1, 18, 22, 1, 30, 1, 1, 42, 46, 1, 58, 1, 66, 70, 1, 78, 82, 1, 1, 1, 102, 106, 1, 1, 126, 130, 1, 138, 1, 150, 1, 162, 166, 1, 178, 1, 190, 1, 1, 198, 210, 222, 226, 1, 1, 238, 1, 250, 1, 262, 1, 270, 1, 1, 282, 1, 306, 310, 1, 1, 330, 1, 346, 1, 1, 358, 366, 1

Offset: 1

Jonathan Sondow, May 14 2010

a(n) == (-1)^((p-1)/2) (mod p), if p = prime(n) is odd.

a(n)*A163366(n) == -1 (mod prime(n)), by Wilson's theorem.

			a(1) = 1 = the empty product, because there are no quadratic nonresidues of prime(1) = 2.
a(4) = 6 because the quadratic nonresidues of prime(4) = 7 are 3, 5, and 6, and 3*5*6 = 90 == 6 (mod 7).

Carl-Erik Froeberg, On sums and products of quadratic residues, BIT, Nord. Tidskr. Inf.-behandl. 11 (1971) 389-398.

Rahul Gupta, Algorithmic Number Theory, Section 24.5

Cf. A177861 Product of the quadratic nonresidues of prime(n), A163366 Product modulo p of the quadratic residues of p, where p = prime(n).

Table[Mod[ Apply[Times, Flatten[Position[ Table[JacobiSymbol[i, Prime[n]], {i, 1, Prime[n] - 1}], -1]]], Prime[n]], {n, 1, 80}]

a(n) = A177861(n) modulo prime(n).

cp's OEIS Frontend

A177860 Product of the quadratic residues of prime(n).

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