A321108 Digits of one of the three 13-adic integers 5^(1/3) that is related to A321105.

11, 11, 5, 11, 2, 0, 9, 0, 6, 11, 9, 6, 7, 9, 2, 9, 9, 2, 3, 3, 8, 2, 7, 11, 6, 7, 4, 7, 10, 5, 5, 4, 11, 6, 2, 5, 2, 7, 10, 9, 9, 2, 9, 5, 7, 7, 4, 5, 10, 4, 1, 6, 4, 1, 4, 0, 4, 10, 11, 4, 12, 12, 7, 2, 9, 6, 11, 8, 5, 6, 11, 2, 0, 6, 6, 12, 10, 8, 12, 11, 2

Offset: 0

Author

Jianing Song, Aug 27 2019

Keywords

• nonn
• base

Comments

For k not divisible by 5, k is a cube in 13-adic field if and only if k == 1, 5, 8, 12 (mod 13). If k is a cube in 13-adic field, then k has exactly three cubic roots.

Examples

			The unique number k in [1, 13^3] and congruent to 11 modulo 13 such that k^3 - 5 is divisible by 13^3 is k = 999 = (5BB)_13, so the first three terms are 11, 11 and 5.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Wikipedia, p-adic number

Crossrefs

Cf. A320914, A320915, A321105, A321106, A321107.

For 5-adic cubic roots, see A290566, A290563, A309443.

Programs

PARI

a(n) = lift(sqrtn(5+O(13^(n+1)), 3) * (-1-sqrt(-3+O(13^(n+1))))/2)\13^n

Formula

a(n) = (A321105(n+1) - A321105(n))/13^n.