A320914 One of the three successive approximations up to 13^n for 13-adic integer 5^(1/3). This is the 7 (mod 13) case (except for n = 0).

0, 7, 7, 1021, 20794, 77916, 4533432, 57628331, 810610535, 8967917745, 40781415864, 592215383260, 22098140111704, 208482821091552, 3842984100198588, 23529866028695033, 586574689183693360, 5244490953465952247, 74447818308516655711, 524269446116346228227, 9295791188369022892289

Offset

0,2

Comments

For n > 0, a(n) is the unique number k in [1, 13^n] and congruent to 7 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.

Links

Table of n, a(n) for n=0..20.

Wikipedia, p-adic number

Example

The unique number k in [1, 13^2] and congruent to 7 modulo 13 such that k^3 - 5 is divisible by 13^2 is k = 7, so a(2) = 7.
The unique number k in [1, 13^3] and congruent to 7 modulo 13 such that k^3 - 5 is divisible by 13^3 is k = 1021, so a(3) = 1021.

Prog

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

Crossrefs

Cf. A320915, A321105, A321106, A321107, A321108.

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

Sequence in context: A180321 A213150 A239152 * A155846 A221114 A198841

Adjacent sequences: A320911 A320912 A320913 * A320915 A320916 A320917

Keyword

nonn

Author

Jianing Song, Aug 27 2019

Status

approved