A320915 One of the three successive approximations up to 13^n for 13-adic integer 5^(1/3). This is the 8 (mod 13) case (except for n = 0).

0, 8, 8, 177, 11162, 211089, 211089, 24345134, 777327338, 7303173106, 113348166836, 1629791577175, 12382753941397, 222065520043726, 1130690839820485, 16880196382617641, 272809661453071426, 5596142534918510154, 14246558454299848087, 576523593214086813732, 4962284464340425145763

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 8 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 8 modulo 13 such that k^3 - 5 is divisible by 13^2 is k = 8, so a(2) = 8.
The unique number k in [1, 13^3] and congruent to 8 modulo 13 such that k^3 - 5 is divisible by 13^3 is k = 177, so a(3) = 177.
		

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A320914, A321105, A321106, A321107, A321108.

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

Programs

PARI

a(n) = lift(sqrtn(5+O(13^n), 3))