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A322085
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One of the two successive approximations up to 13^n for 13-adic integer sqrt(3). Here the 4 (mod 13) case (except for n = 0).
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9
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0, 4, 108, 1122, 18698, 361430, 1104016, 5930825, 570667478, 7912243967, 113957237697, 251815729546, 11004778093768, 104197118583692, 3132948184506222, 26757206498701956, 589802029653700283, 7909384730668678534, 85763128005100719931, 648040162764887685576
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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 solution to x^2 == 3 (mod 13^n) in the range [0, 13^n - 1] and congruent to 4 modulo 13.
A322086 is the approximation (congruent to 9 mod 13) of another square root of 3 over the 13-adic field.
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LINKS
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FORMULA
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For n > 0, a(n) = 13^n - A322086(n).
a(n) = Sum_{i=0..n-1} A322087(i)*13^i.
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EXAMPLE
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4^2 = 16 = 1*13 + 3.
108^2 = 11664 = 69*13^2 + 3.
1122^2 = 1258884 = 573*13^3 + 3.
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MAPLE
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S:= map(t -> op([1, 3], t), [padic:-evalp(RootOf(x^2-3, x), 13, 30)]):
S4:= op(select(t -> t[1]=4, S)):
seq(add(S4[i]*13^(i-1), i=1..n-1), n=1..31); # Robert Israel, Jun 13 2019
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PROG
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(PARI) a(n) = truncate(sqrt(3+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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