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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).
9
0, 1, 21, 121, 246, 2121, 5246, 52121, 286496, 677121, 677121, 20208371, 117864621, 606145871, 3047552121, 3047552121, 94600286496, 704951848996, 2993770208371, 2993770208371, 79287715520871, 270022578802121, 746859737005246, 5515231319036496, 29357089229192746
OFFSET
0,3
COMMENTS
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.
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.
FORMULA
a(n) = A325485(n)*A048899(n) mod 5^n = A325486(n)*A048898(n) mod 5^n.
For n > 0, a(n) = 5^n - A325487(n).
a(n)^2 == A324023(n) (mod 5^n).
EXAMPLE
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.
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.
PROG
(PARI) a(n) = lift(sqrtn(6+O(5^n), 4))
CROSSREFS
Approximations of p-adic fourth-power roots:
this sequence, A325485, A325486, A325487 (5-adic, 6^(1/4));
A324077, A324082, A324083, A324084 (13-adic, 3^(1/4)).
Sequence in context: A316713 A044353 A044734 * A365205 A361699 A200888
KEYWORD
nonn
AUTHOR
Jianing Song, Sep 07 2019
STATUS
approved