A342137 Least k such that k^8 - 16 has a root modulo prime(n) where n is the sequence index.

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

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

Frank M Jackson, Table of n, a(n) for n = 1..10000

Wikipedia, Grunwald-Wang theorem.

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

Adjacent sequences: A342134 A342135 A342136 * A342138 A342139 A342140

Keyword

nonn

Author

Frank M Jackson, Mar 01 2021

Status

approved