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Digits of one of the two 17-adic integers sqrt(-2) that is related to A322564.
6

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

%S 10,15,4,16,7,16,0,8,11,0,2,16,15,1,5,0,1,8,3,1,5,11,5,13,7,0,0,1,13,

%T 13,16,1,9,1,0,13,2,7,4,11,14,14,12,4,4,5,5,16,7,1,4,14,7,2,14,6,10,

%U 16,8,11,1,10,10,2,7,14,6,15,9,14,3,4,13,3,10,0

%N Digits of one of the two 17-adic integers sqrt(-2) that is related to A322564.

%C This square root of -2 in the 17-adic field ends with digit 10 (A when written as a 17-adic number). The other, A322565, ends with digit 7.

%H Seiichi Manyama, <a href="/A322566/b322566.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(n) = (A322564(n+1) - A322564(n))/17^n.

%F For n > 0, a(n) = 16 - A322565(n).

%F Equals A309989*A322562 = A309990*A322561.

%e The solution to x^2 == -2 (mod 17^4) such that x == 10 (mod 17) is x == 80029 (mod 17^4), and 80029 is written as G4FA in heptadecimal, so the first four terms are 10, 15, 4 and 16.

%o (PARI) a(n) = truncate(-sqrt(-2+O(17^(n+1))))\17^n

%Y Cf. A322563, A322564.

%Y Digits of 17-adic square roots:

%Y A309989, A309990 (sqrt(-1));

%Y A322561, A322562 (sqrt(2));

%Y A322565, this sequence (sqrt(-2)).

%K nonn,base

%O 0,1

%A _Jianing Song_, Aug 29 2019