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A322559
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).
5
0, 6, 244, 4290, 43594, 461199, 6140627, 344066593, 6088808015, 54919110102, 292094863096, 30532003369831, 544610447984326, 6953455057511697, 56476345222041382, 1066743304578446956, 21103704665147157507, 118426088416480894469, 11699789754825195592947
OFFSET
0,2
COMMENTS
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.
A322560 is the approximation (congruent to 11 mod 17) of another square root of 2 over the 17-adic field.
FORMULA
For n > 0, a(n) = 17^n - A322560(n).
a(n) = Sum_{i=0..n-1} A322561(i)*17^i.
a(n) = A286877(n)*A322564(n) mod 17^n = A286878(n)*A322563(n) mod 17^n.
EXAMPLE
6^2 = 36 = 2*17 + 2;
244^2 = 59536 = 206*17^2 + 2;
4290^2 = 18404100 = 3746*17^3 + 2.
PROG
(PARI) a(n) = truncate(sqrt(2+O(17^n)))
CROSSREFS
Approximations of 17-adic square roots:
A286877, A286878 (sqrt(-1));
this sequence, A322560 (sqrt(2));
A322563, A322564 (sqrt(-2)).
Sequence in context: A231019 A254009 A072228 * A229631 A206307 A229475
KEYWORD
nonn
AUTHOR
Jianing Song, Aug 29 2019
STATUS
approved