A373748 Triangle read by rows: T(n, k) is k if k is a quadratic residue modulo n, otherwise is -k and is a quadratic nonresidue modulo n. T(0, 0) = 0 by convention.

0, 0, 1, 0, 1, 2, 0, 1, -2, 3, 0, 1, -2, -3, 4, 0, 1, -2, -3, 4, 5, 0, 1, -2, 3, 4, -5, 6, 0, 1, 2, -3, 4, -5, -6, 7, 0, 1, -2, -3, 4, -5, -6, -7, 8, 0, 1, -2, -3, 4, -5, -6, 7, -8, 9, 0, 1, -2, -3, 4, 5, 6, -7, -8, 9, 10, 0, 1, -2, 3, 4, 5, -6, -7, -8, 9, -10, 11, 0, 1, -2, -3, 4, -5, -6, -7, -8, 9, -10, -11, 12

Offset: 0

Author

Peter Luschny, Jun 27 2024

Keywords

• sign
• tabl

Examples

			Triangle starts:
  [0] [0]
  [1] [0,  1]
  [2] [0,  1,  2]
  [3] [0,  1, -2,  3]
  [4] [0,  1, -2, -3,  4]
  [5] [0,  1, -2, -3,  4,  5]
  [6] [0,  1, -2,  3,  4, -5,  6]
  [7] [0,  1,  2, -3,  4, -5, -6,  7]
  [8] [0,  1, -2, -3,  4, -5, -6, -7,  8]
  [9] [0,  1, -2, -3,  4, -5, -6,  7, -8,  9]
 [10] [0,  1, -2, -3,  4,  5,  6, -7, -8,  9,  10]
		

Links

• Carl Friedrich Gauss, Vierter Abschnitt. Von den Congruenzen zweiten Grades. Quadratische Reste und Nichtreste. Art. 97, in "Untersuchungen über die höhere Arithmetik", Hrsg. H. Maser, Verlag von Julius Springer, Berlin, 1889.
• Peter Luschny, SageMath: is_quadratic_residue.

Crossrefs

Signed version of A002262.

Cf. A000004 (column 0), A001477 (main diagonal), A255644(n) + n (row sums).

Cf. A096008, A096013, A373749.

Programs

Maple

QR := (a, n) -> ifelse(n = 0, 1, NumberTheory:-QuadraticResidue(a, n)):
for n from 0 to 10 do seq(a*QR(a, n), a = 0..n) od;

Mathematica

qr[n_] := qr[n] = Join[Table[PowerMod[k, 2, n], {k, 0, Floor[n/2]}], {n}];
T[0, 0] := 0; T[n_, k_] := If[MemberQ[qr[n], k], k, -k];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten

SageMath

def Trow(n):
    q = set(mod(a * a, n) for a in range(n // 2  + 1)).union({n})
    return [k if k in q else -k for k in range(n + 1)]
for n in range(11): print(Trow(n))