A322560 - OEIS

%I #11 Aug 29 2019 10:41:26

%S 0,11,45,623,39927,958658,17996942,66272080,886949426,63668766395,

%T 1723899037353,3739892937802,38011789245435,2951122975394240,

%U 111901481337359547,1795679746931368837,27557487210519710974,708814173469855869708,2363294697242529398062

%N One of the two successive approximations up to 17^n for 17-adic integer sqrt(2). This is the 11 (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 11 modulo 17.

%C A322559 is the approximation (congruent to 6 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 - A322559(n).

%F a(n) = Sum_{i=0..n-1} A322562(i)*17^i.

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

%e 11^2 = 121 = 7*17 + 2;

%e 45^2 = 2025 = 7*17^2 + 2;

%e 623^2 = 388129 = 79*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 A322559, this sequence (sqrt(2));

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

%K nonn

%O 0,2

%A _Jianing Song_, Aug 29 2019