A324153 Digits of one of the four 13-adic integers 3^(1/4) that is congruent to 11 mod 13.

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

Offset: 0

Author

Jianing Song, Sep 01 2019

Keywords

• nonn
• base

Comments

One of the two square roots of A322087, where an A-number represents a 13-adic number. The other square root is A324085.

For k not divisible by 13, k is a fourth power in 13-adic field if and only if k == 1, 3, 9 (mod 13). If k is a fourth power in 13-adic field, then k has exactly 4 fourth-power roots.

Examples

			The unique number k in [1, 13^3] and congruent to 11 modulo 13 such that k^4 - 3 is divisible by 13^3 is k = 986 = (5AB)_13, so the first three terms are 11, 10 and 5.
		

Links

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

Crossrefs

Cf. A286838, A286839, A322087, A324077, A324082, A324083, A324084, A324085, A324086, A324087.

Programs

Maple

R:= select(t -> op([1,3,1],t)=11, [padic:-rootp(x^4-3, 13,101)]):
op([1,1,3],R); # Robert Israel, Sep 08 2019

PARI

a(n) = lift(-sqrtn(3+O(13^(n+1)), 4) * sqrt(-1+O(13^(n+1))))\13^n

Formula

Equals A324086*A286839 = A324087*A286838.

a(n) = (A324084(n+1) - A324084(n))/13^n.

For n > 0, a(n) = 12 - A324085(n).