A322563 One of the two successive approximations up to 17^n for 17-adic integer sqrt(-2). This is the 7 (mod 17) case (except for n = 0).

0, 7, 24, 3492, 3492, 755181, 755181, 386956285, 3669665669, 38548452874, 1935954476826, 30159869083112, 30159869083112, 612782106312873, 149181452599901928, 2001337544753312147, 47800106368910364835, 777717984503913392050, 7395640079594607505466

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 7 modulo 17.

A322564 is the approximation (congruent to 10 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 - A322564(n).

a(n) = Sum_{i=0..n-1} A322565(i)*17^i.

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

Example

7^2 = 49 = 3*17 - 2;
24^2 = 576 = 2*17^2 - 2;
3492^2 = 12194064 = 2482*17^3 - 2.

Mathematica

{0}~Join~Table[First@Select[PowerModList[-2, 1/2, 17^k], Mod[#, 17]==7&], {k, 20}] (* Giorgos Kalogeropoulos, Sep 14 2022 *)

Prog

(PARI) a(n) = truncate(sqrt(-2+O(17^n)))

Crossrefs

Cf. A322565, A322566.

Approximations of 17-adic square roots:

A286877, A286878 (sqrt(-1));

A322559, A322560 (sqrt(2));

this sequence, A322564 (sqrt(-2)).

Sequence in context: A012777 A074783 A286742 * A065658 A242322 A249437

Adjacent sequences: A322560 A322561 A322562 * A322564 A322565 A322566

Keyword

nonn

Author

Jianing Song, Aug 29 2019

Status

approved