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

Author

Jonathan Sondow, May 14 2010

Keywords

• easy
• nonn

Comments

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

Examples

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

References

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

Links

• Rahul Gupta, Algorithmic Number Theory, Section 24.5

Crossrefs

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).

Programs

Mathematica

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

Formula

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