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A324084
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One of the four successive approximations up to 13^n for 13-adic integer 3^(1/4).This is the 11 (mod 13) case (except for n = 0).
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13
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0, 11, 141, 986, 25153, 25153, 2252911, 2252911, 504241047, 3767163931, 67394160169, 1583837570508, 5168158358582, 191552839338430, 2008803478891948, 21695685407388393, 226439257463751421, 1557272475830111103, 96711847589024828366, 96711847589024828366
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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, 13^n] and congruent to 11 mod 13 such that k^4 - 3 is divisible by 13^n.
For k not divisible by 13, k is a fourth power in 13-adic field if and only if k == 1, 3, 9 (mod 13). If k is a fourth power in 13-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) = 13^n - A324077(n).
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EXAMPLE
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The unique number k in [1, 13^2] and congruent to 11 modulo 13 such that k^4 - 3 is divisible by 13^2 is k = 141, so a(2) = 141.
The unique number k in [1, 13^3] and congruent to 11 modulo 13 such that k^4 - 3 is divisible by 13^3 is k = 986, so a(3) = 986.
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PROG
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(PARI) a(n) = lift(-sqrtn(3+O(13^n), 4) * sqrt(-1+O(13^n)))
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CROSSREFS
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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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