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A342137 Least k such that k^8 - 16 has a root modulo prime(n) where n is the sequence index. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
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.
LINKS
EXAMPLE
a(4) = 3 because prime(4) = 7 and the least k such that k^8 - 16 = 0 (mod 7) is k = 3.
MATHEMATICA
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
PROG
(PARI) a(n) = my(k=0, p=prime(n)); while(Mod(k, p)^8 != 16, k++); k; \\ Michel Marcus, Mar 01 2021
CROSSREFS
Sequence in context: A163523 A151664 A083503 * A062069 A163375 A027011
KEYWORD
nonn
AUTHOR
Frank M Jackson, Mar 01 2021
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)