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%I #20 Mar 04 2021 11:05:46
%S 0,1,1,3,3,4,3,6,5,11,8,5,8,16,7,22,23,10,20,12,12,9,9,25,14,9,38,31,
%T 32,14,16,28,31,50,43,46,27,18,13,79,78,18,57,34,13,20,93,15,15,106,
%U 85,99,22,91,15,110,81,96,59,29,127,137,108,66,24,113,75,26,107
%N Least k such that k^8 - 16 has a root modulo prime(n) where n is the sequence index.
%C This sequence of least roots modulo prime(n) for k^8 - 16, represents the counterexample to a conjecture that stated if k^m - j has a root for modulo every prime, then j must be of the form j = y^m. The corrected theorem which excludes the counterexample is known as the Grunwald-Wang Theorem.
%C Note that a(n) < prime(n)/2 for all n.
%H Frank M Jackson, <a href="/A342137/b342137.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Grunwald-Wang_theorem">Grunwald-Wang theorem</a>.
%e a(4) = 3 because prime(4) = 7 and the least k such that k^8 - 16 = 0 (mod 7) is k = 3.
%t lst = {}; Do[Do[If[Mod[m^8, Prime[n]]==Mod[16, Prime[n]], AppendTo[lst, m]; Break[]], {m, 0, Prime[n]-1}], {n, 1, 100}]; lst
%o (PARI) a(n) = my(k=0, p=prime(n)); while(Mod(k, p)^8 != 16, k++); k; \\ _Michel Marcus_, Mar 01 2021
%K nonn
%O 1,4
%A _Frank M Jackson_, Mar 01 2021
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