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A325484 One of the four successive approximations up to 5^n for the 5-adic integer 6^(1/4). This is the 1 (mod 5) case (except for n = 0).

%I A325484 #7 Sep 11 2019 20:33:37
%S A325484 0,1,21,121,246,2121,5246,52121,286496,677121,677121,20208371,
%T A325484 117864621,606145871,3047552121,3047552121,94600286496,704951848996,
%U A325484 2993770208371,2993770208371,79287715520871,270022578802121,746859737005246,5515231319036496,29357089229192746
%N A325484 One of the four successive approximations up to 5^n for the 5-adic integer 6^(1/4). This is the 1 (mod 5) case (except for n = 0).
%C A325484 For n > 0, a(n) is the unique number k in [1, 5^n] and congruent to 1 mod 5 such that k^4 - 6 is divisible by 5^n.
%C A325484 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.
%H A325484 Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">p-adic number</a>
%F A325484 a(n) = A325485(n)*A048899(n) mod 5^n = A325486(n)*A048898(n) mod 5^n.
%F A325484 For n > 0, a(n) = 5^n - A325487(n).
%F A325484 a(n)^2 == A324023(n) (mod 5^n).
%e A325484 The unique number k in [1, 5^2] and congruent to 1 modulo 5 such that k^4 - 6 is divisible by 5^2 is k = 21, so a(2) = 21.
%e A325484 The unique number k in [1, 5^3] and congruent to 1 modulo 5 such that k^4 - 6 is divisible by 5^3 is k = 121, so a(3) = 121.
%o A325484 (PARI) a(n) = lift(sqrtn(6+O(5^n), 4))
%Y A325484 Cf. A048898, A048899, A324023, A325489, A325490, A325491, A325492.
%Y A325484 Approximations of p-adic fourth-power roots:
%Y A325484 this sequence, A325485, A325486, A325487 (5-adic, 6^(1/4));
%Y A325484 A324077, A324082, A324083, A324084 (13-adic, 3^(1/4)).
%K A325484 nonn
%O A325484 0,3
%A A325484 _Jianing Song_, Sep 07 2019