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A325487
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One of the four successive approximations up to 13^n for the 13-adic integer 6^(1/4). This is the 4 (mod 5) case (except for n = 0).
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9
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0, 4, 4, 4, 379, 1004, 10379, 26004, 104129, 1276004, 9088504, 28619754, 126276004, 614557254, 3055963504, 27470026004, 57987604129, 57987604129, 820927057254, 16079716119754, 16079716119754, 206814579401004, 1637326054010379, 6405697636041629, 30247555546197879
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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 4 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 - A325484(n).
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EXAMPLE
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The unique number k in [1, 5^2] and congruent to 4 modulo 5 such that k^4 - 6 is divisible by 5^2 is k = 4, so a(2) = 4.
The unique number k in [1, 5^3] and congruent to 4 modulo 5 such that k^4 - 6 is divisible by 5^3 is also k = 4, so a(3) is also 4.
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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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