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A325486
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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).
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9
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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
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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For n > 0, a(n) = 5^n - A325485(n).
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EXAMPLE
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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.
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PROG
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(PARI) a(n) = lift(-sqrtn(6+O(5^n), 4) * sqrt(-1+O(5^n)))
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CROSSREFS
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Approximations of p-adic fourth-power roots:
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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