A322559 - OEIS

%I #11 Aug 29 2019 10:41:13

%S 0,6,244,4290,43594,461199,6140627,344066593,6088808015,54919110102,

%T 292094863096,30532003369831,544610447984326,6953455057511697,

%U 56476345222041382,1066743304578446956,21103704665147157507,118426088416480894469,11699789754825195592947

%N One of the two successive approximations up to 17^n for 17-adic integer sqrt(2). This is the 6 (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 6 modulo 17.

%C A322560 is the approximation (congruent to 11 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 - A322560(n).

%F a(n) = Sum_{i=0..n-1} A322561(i)*17^i.

%F a(n) = A286877(n)*A322564(n) mod 17^n = A286878(n)*A322563(n) mod 17^n.

%e 6^2 = 36 = 2*17 + 2;

%e 244^2 = 59536 = 206*17^2 + 2;

%e 4290^2 = 18404100 = 3746*17^3 + 2.

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

%Y Cf. A322561, A322562.

%Y Approximations of 17-adic square roots:

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

%Y this sequence, A322560 (sqrt(2));

%Y A322563, A322564 (sqrt(-2)).

%K nonn

%O 0,2

%A _Jianing Song_, Aug 29 2019