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A322090
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One of the two successive approximations up to 13^n for 13-adic integer sqrt(3). Here the 7 (mod 13) case (except for n = 0).
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4
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0, 7, 124, 124, 13306, 70428, 1926893, 40541365, 542529501, 2989721664, 45407719156, 458983194703, 18380587135073, 111572927624997, 2231698673770768, 2231698673770768, 462904735800587581, 5120821000082846468, 74324148355133549932, 1423789031778622267480, 10195310774031298931542
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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 7 modulo 13.
A322089 is the approximation (congruent to 6 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 - A322089(n).
a(n) = Sum_{i=0..n-1} A322092(i)*13^i.
a(n) == L(13^n,7) (mod 13^n) == ((7 + sqrt(53))/2)^(13^n) + ((7 - sqrt(53))/2)^(13^n) (mod 13^n), where L(n,x) denotes the n-th Lucas polynomial, the n-th row polynomial of A114525. - Peter Bala, Dec 05 2022
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EXAMPLE
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7^2 = 49 = 4*13 - 3.
124^2 = 15376 = 91*13^2 - 3 = 7*13^3 - 3.
13306^2 = 177049636 = 6199*13^4 - 3.
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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,easy
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AUTHOR
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STATUS
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approved
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