A325487 One of the four successive approximations up to 13^n for the 13-adic integer 6^(1/4). This is the 4 (mod 5) case (except for n = 0).

0, 4, 4, 4, 379, 1004, 10379, 26004, 104129, 1276004, 9088504, 28619754, 126276004, 614557254, 3055963504, 27470026004, 57987604129, 57987604129, 820927057254, 16079716119754, 16079716119754, 206814579401004, 1637326054010379, 6405697636041629, 30247555546197879

Offset: 0

Author

Jianing Song, Sep 07 2019

Keywords

• nonn

Comments

For n > 0, a(n) is the unique number k in [1, 5^n] and congruent to 4 mod 5 such that k^4 - 6 is divisible by 5^n.

For k not divisible by 5, k is a fourth power in 5-adic field if and only if k == 1 (mod 5). If k is a fourth power in 5-adic field, then k has exactly 4 fourth-power roots.

Examples

			The unique number k in [1, 5^2] and congruent to 4 modulo 5 such that k^4 - 6 is divisible by 5^2 is k = 4, so a(2) = 4.
The unique number k in [1, 5^3] and congruent to 4 modulo 5 such that k^4 - 6 is divisible by 5^3 is also k = 4, so a(3) is also 4.
		

Links

• Wikipedia, p-adic number

Crossrefs

Cf. A048898, A048899, A324023, A325489, A325490, A325491, A325492.

Approximations of p-adic fourth-power roots:

A325484, A325485, A325486, this sequence (5-adic, 6^(1/4));

A324077, A324082, A324083, A324084 (13-adic, 3^(1/4)).

Programs

PARI

a(n) = lift(-sqrtn(6+O(5^n), 4))

Formula

a(n) = A325485(n)*A048898(n) mod 5^n = A325486(n)*A048899(n) mod 5^n.

For n > 0, a(n) = 5^n - A325484(n).

a(n)^2 == A324023(n) (mod 5^n).