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%I #28 Jul 09 2024 17:39:10
%S 0,1,1,-1,-1,1,0,-1,-1,1,1,1,1,1,1,-1,0,-1,1,-1,-1,0,1,-1,-1,-1,-1,1,
%T 1,-1,0,-1,-1,-1,1,-1,-1,-1,1,-1,-1,1,-1,1,-1,0,1,1,1,1,1,1,1,1,1,1,0,
%U -1,-1,1,-1,-1,-1,-1,1,1,-1,1,1,0,-1,-1,1,-1,-1,-1,1,-1
%N Legendre's triangle read by rows. T(n, k) = L(n / prime(k)) where L(n/p) is the Legendre symbol, for n >= 0 and 2 <= k <= n + 2.
%H Paolo Xausa, <a href="/A372726/b372726.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of the triangle, flattened).
%H Adrien Marie Legendre, <a href="https://www.e-rara.ch/zut/content/structure/1089237">Essai sur la théorie des nombres</a>, Paris, Duprat, an VI [1798]. <a href="https://www.e-rara.ch/zut/content/pageview/1089453">Introducing the symbol</a>, p. 186.
%F T(n, k) = r - p*[r > 1] where r = n^v mod p, p = prime(k), v = (p - 1)/2, and [.] are the Iverson brackets.
%e Triangle starts:
%e [ 0] 0;
%e [ 1] 1, 1;
%e [ 2] -1, -1, 1;
%e [ 3] 0, -1, -1, 1;
%e [ 4] 1, 1, 1, 1, 1;
%e [ 5] -1, 0, -1, 1, -1, -1;
%e [ 6] 0, 1, -1, -1, -1, -1, 1;
%e [ 7] 1, -1, 0, -1, -1, -1, 1, -1;
%e [ 8] -1, -1, 1, -1, -1, 1, -1, 1, -1;
%e [ 9] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1;
%e [10] 1, 0, -1, -1, 1, -1, -1, -1, -1, 1, 1;
%e .
%e Not limiting the range of k leads to the square array:
%e .
%e [n\p] 3, 5, 7, 11, 13, 17, 19, 23, 29, 31
%e -----------------------------------------------
%e [0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e [1] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e [2] -1, -1, 1, -1, -1, 1, -1, 1, -1, 1, ...
%e [3] 0, -1, -1, 1, 1, -1, -1, 1, -1, -1, ...
%e [4] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e [5] -1, 0, -1, 1, -1, -1, 1, -1, 1, 1, ...
%e [6] 0, 1, -1, -1, -1, -1, 1, 1, 1, -1, ...
%e [7] 1, -1, 0, -1, -1, -1, 1, -1, 1, 1, ...
%e [8] -1, -1, 1, -1, -1, 1, -1, 1, -1, 1, ...
%e [9] 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e ...
%p L := (n, k) -> NumberTheory:-LegendreSymbol(n, ithprime(k)):
%p for n from 0 to 10 do lprint([n], seq(L(n, k), k = 2..n + 2)) od;
%t Array[JacobiSymbol[#, Prime[Range[2, #+2]]]&, 15, 0] (* _Paolo Xausa_, Jul 09 2024 *)
%o (Python)
%o from sympy import primerange, prime, legendre_symbol
%o for n in range(11):
%o print([n], [legendre_symbol(n, p) for p in primerange(3, prime(n + 2) + 1)])
%o # For illustration of the formula (Sympy's implementation is more efficent):
%o def LegendreSymbol(n, p):
%o v = (p - 1) // 2
%o res = pow(n, v, p)
%o return res - p if res > 1 else res
%Y Family: A217831 (Euclid's triangle), A372877 (Jacobi's triangle), A372728 (Kronecker's triangle), A373223 (Gauss' triangle), A373751 (quadratic residue modulo prime(n)), A373748 (quadratic residue/nonresidue modulo n).
%Y Cf. A372725 (row sums).
%K sign,tabl
%O 0
%A _Peter Luschny_, May 22 2024
%E Data corrected by _Paolo Xausa_, Jul 09 2024
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