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).

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.

Links

Table of n, a(n) for n=0..18.

Wikipedia, p-adic number

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

Cf. A322561, A322562.

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

Adjacent sequences: A322556 A322557 A322558 * A322560 A322561 A322562

Keyword

nonn

Author

Jianing Song, Aug 29 2019

Status

approved