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 A322563 One of the two successive approximations up to 17^n for 17-adic integer sqrt(-2). This is the 7 (mod 17) case (except for n = 0). 5
 0, 7, 24, 3492, 3492, 755181, 755181, 386956285, 3669665669, 38548452874, 1935954476826, 30159869083112, 30159869083112, 612782106312873, 149181452599901928, 2001337544753312147, 47800106368910364835, 777717984503913392050, 7395640079594607505466 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS For n > 0, a(n) is the unique solution to x^2 == -2 (mod 17^n) in the range [0, 17^n - 1] and congruent to 7 modulo 17. A322564 is the approximation (congruent to 10 mod 17) of another square root of -2 over the 17-adic field. LINKS Table of n, a(n) for n=0..18. Wikipedia, p-adic number FORMULA For n > 0, a(n) = 17^n - A322564(n). a(n) = Sum_{i=0..n-1} A322565(i)*17^i. a(n) = A286877(n)*A322559(n) mod 17^n = A286878(n)*A322560(n) mod 17^n. EXAMPLE 7^2 = 49 = 3*17 - 2; 24^2 = 576 = 2*17^2 - 2; 3492^2 = 12194064 = 2482*17^3 - 2. MATHEMATICA {0}~Join~Table[First@Select[PowerModList[-2, 1/2, 17^k], Mod[#, 17]==7&], {k, 20}] (* Giorgos Kalogeropoulos, Sep 14 2022 *) PROG (PARI) a(n) = truncate(sqrt(-2+O(17^n))) CROSSREFS Cf. A322565, A322566. Approximations of 17-adic square roots: A286877, A286878 (sqrt(-1)); A322559, A322560 (sqrt(2)); this sequence, A322564 (sqrt(-2)). Sequence in context: A012777 A074783 A286742 * A065658 A242322 A249437 Adjacent sequences: A322560 A322561 A322562 * A322564 A322565 A322566 KEYWORD nonn AUTHOR Jianing Song, Aug 29 2019 STATUS approved

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Last modified October 3 09:00 EDT 2023. Contains 365854 sequences. (Running on oeis4.)