A322564 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).

0, 10, 265, 1421, 80029, 664676, 23382388, 23382388, 3306091772, 80039423623, 80039423623, 4112027224521, 552462368146649, 9291795926593064, 19196373959499001, 861085506756503646, 861085506756503646, 49522277382423372127, 6667444372473117485543, 48856697728676292458570, 287929133413827617305723

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 10 modulo 17.

A322563 is the approximation (congruent to 7 mod 17) of another square root of -2 over the 17-adic field.

Links

Table of n, a(n) for n=0..20.

Wikipedia, p-adic number

Formula

For n > 0, a(n) = 17^n - A322563(n).

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

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

Example

10^2 = 100 = 6*17 - 2;
265^2 = 70225 = 243*17^2 - 2;
1421^2 = 2019241 = 411*17^3 - 2.

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));

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

Sequence in context: A160481 A060608 A003388 * A055408 A245983 A336027

Adjacent sequences: A322561 A322562 A322563 * A322565 A322566 A322567

Keyword

nonn

Author

Jianing Song, Aug 29 2019

Status

approved