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Note that GCD[m, n] yields a more complicated pattern (see page 613 ), as do JacobiSymbol[m, 2n - 1] (see page 1081 ) and various combinations of functions (see page 747 ).

It is also flat for maximal length LFSR sequences (see page 1084 ) and for sequences JacobiSymbol[Range[0, p - 1], p] with prime p (see page 870 ). … If Mod[p, 4] 1 JacobiSymbol sequences also satisfy Fourier[list] list .

The so-called Paley family of Hadamard matrices for n = 4k = p + 1 with p prime are given by
PadLeft[Array[JacobiSymbol[#2 - #1, n - 1]&, {n, n} - 1] - IdentityMatrix[n - 1], {n, n}, 1]
Originally introduced by Jacques Hadamard in 1893 as the matrices with elements Abs[a] ≤ 1 which attain the maximal possible determinant ± n n/2 , Hadamard matrices appear in various combinatorial problems, particularly design of exhaustive combinations of experiments and Reed–Muller error-correcting codes.

If m is a prime p , then the simple tests JacobiSymbol[x, p] 1 (see page 1081 ) or Mod[x (p - 1)/2 , p] 1 determine whether x is a quadratic residue.