A325486 - OEIS

A325486

One of the four successive approximations up to 5^n for the 5-adic integer 6^(1/4). This is the 3 (mod 5) case (except for n = 0).

0, 3, 3, 103, 228, 2728, 8978, 71478, 71478, 1633978, 3587103, 42649603, 140305853, 628587103, 3069993353, 21380540228, 82415696478, 540179368353, 540179368353, 15798968430853, 34872454758978, 34872454758978, 988546771165228, 8141104144212103, 8141104144212103

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OFFSET

0,2

COMMENTS

For n > 0, a(n) is the unique number k in [1, 5^n] and congruent to 3 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.

LINKS

Table of n, a(n) for n=0..24.

Wikipedia, p-adic number

FORMULA

a(n) = A325484(n)*A048899(n) mod 13^n = A325485(n)*A048898(n) mod 13^n.

For n > 0, a(n) = 5^n - A325485(n).

a(n)^2 == A324024(n) (mod 5^n).

EXAMPLE

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

The unique number k in [1, 5^3] and congruent to 3 modulo 5 such that k^4 - 6 is divisible by 5^3 is k = 103, so a(3) = 103.

PROG

(PARI) a(n) = lift(-sqrtn(6+O(5^n), 4) * sqrt(-1+O(5^n)))