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A325484
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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 1 (mod 5) case (except for n = 0).
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9
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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
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OFFSET
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0,3
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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 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.
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LINKS
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FORMULA
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For n > 0, a(n) = 5^n - A325487(n).
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EXAMPLE
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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.
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PROG
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(PARI) a(n) = lift(sqrtn(6+O(5^n), 4))
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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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