A321105 One of the three successive approximations up to 13^n for 13-adic integer 5^(1/3). This is the 11 (mod 13) case (except for n = 0).

0, 11, 154, 999, 25166, 82288, 82288, 43523569, 43523569, 4937907895, 121587400998, 1362313827639, 12115276191861, 175201872049228, 2901077831379505, 10775830602778083, 471448867729594896, 6460198350378213465, 23761030189140889331, 361127251045013068718, 4746888122171351400749

Offset: 0

Author

Jianing Song, Aug 27 2019

Keywords

• nonn

Comments

For n > 0, a(n) is the unique number k in [1, 13^n] and congruent to 11 mod 13 such that k^3 - 5 is divisible by 13^n.

For k not divisible by 13, 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, 13^2] and congruent to 11 modulo 13 such that k^3 - 5 is divisible by 13^2 is k = 154, so a(2) = 154.
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, so a(3) = 999.
		

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A320914, A320915, A321106, A321107, A321108.

For 5-adic cubic roots, see A290567, A290568, A309444.

Programs

PARI

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