A322564 - OEIS

%I #14 Mar 26 2022 17:46:37

%S 0,10,265,1421,80029,664676,23382388,23382388,3306091772,80039423623,

%T 80039423623,4112027224521,552462368146649,9291795926593064,

%U 19196373959499001,861085506756503646,861085506756503646,49522277382423372127,6667444372473117485543,48856697728676292458570,287929133413827617305723

%N One of the two successive approximations up to 17^n for 17-adic integer sqrt(-2). This is the 10 (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 10 modulo 17.

%C A322563 is the approximation (congruent to 7 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 - A322563(n).

%F a(n) = Sum_{i=0..n-1} A322566(i)*17^i.

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

%e 10^2 = 100 = 6*17 - 2;

%e 265^2 = 70225 = 243*17^2 - 2;

%e 1421^2 = 2019241 = 411*17^3 - 2.

%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 A322563, this sequence (sqrt(-2)).

%K nonn

%O 0,2

%A _Jianing Song_, Aug 29 2019