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A342137
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Least k such that k^8 - 16 has a root modulo prime(n) where n is the sequence index.
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1
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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, 32, 14, 16, 28, 31, 50, 43, 46, 27, 18, 13, 79, 78, 18, 57, 34, 13, 20, 93, 15, 15, 106, 85, 99, 22, 91, 15, 110, 81, 96, 59, 29, 127, 137, 108, 66, 24, 113, 75, 26, 107
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OFFSET
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1,4
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COMMENTS
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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.
Note that a(n) < prime(n)/2 for all n.
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LINKS
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EXAMPLE
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a(4) = 3 because prime(4) = 7 and the least k such that k^8 - 16 = 0 (mod 7) is k = 3.
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MATHEMATICA
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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
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PROG
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(PARI) a(n) = my(k=0, p=prime(n)); while(Mod(k, p)^8 != 16, k++); k; \\ Michel Marcus, Mar 01 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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