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

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

Author

Jonathan Sondow, May 14 2010

Keywords

• easy
• nonn

Comments

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.

Examples

			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).
		

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

Programs

Mathematica

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

Formula

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