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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

%I #18 Oct 12 2022 10:56:22

%S 0,7,24,3492,3492,755181,755181,386956285,3669665669,38548452874,

%T 1935954476826,30159869083112,30159869083112,612782106312873,

%U 149181452599901928,2001337544753312147,47800106368910364835,777717984503913392050,7395640079594607505466

%N 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).

%C 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.

%C A322564 is the approximation (congruent to 10 mod 17) of another square root of -2 over the 17-adic field.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/P-adic_number">p-adic number</a>

%F For n > 0, a(n) = 17^n - A322564(n).

%F a(n) = Sum_{i=0..n-1} A322565(i)*17^i.

%F a(n) = A286877(n)*A322559(n) mod 17^n = A286878(n)*A322560(n) mod 17^n.

%e 7^2 = 49 = 3*17 - 2;

%e 24^2 = 576 = 2*17^2 - 2;

%e 3492^2 = 12194064 = 2482*17^3 - 2.

%t {0}~Join~Table[First@Select[PowerModList[-2,1/2,17^k],Mod[#,17]==7&],{k,20}] (* _Giorgos Kalogeropoulos_, Sep 14 2022 *)

%o (PARI) a(n) = truncate(sqrt(-2+O(17^n)))

%Y Cf. A322565, A322566.

%Y Approximations of 17-adic square roots:

%Y A286877, A286878 (sqrt(-1));

%Y A322559, A322560 (sqrt(2));

%Y this sequence, A322564 (sqrt(-2)).

%K nonn

%O 0,2

%A _Jianing Song_, Aug 29 2019