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A322090 One of the two successive approximations up to 13^n for 13-adic integer sqrt(3). Here the 7 (mod 13) case (except for n = 0). 4
0, 7, 124, 124, 13306, 70428, 1926893, 40541365, 542529501, 2989721664, 45407719156, 458983194703, 18380587135073, 111572927624997, 2231698673770768, 2231698673770768, 462904735800587581, 5120821000082846468, 74324148355133549932, 1423789031778622267480, 10195310774031298931542 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For n > 0, a(n) is the unique solution to x^2 == -3 (mod 13^n) in the range [0, 13^n - 1] and congruent to 7 modulo 13.

A322089 is the approximation (congruent to 6 mod 13) of another square root of -3 over the 13-adic field.

LINKS

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

Wikipedia, p-adic number

FORMULA

For n > 0, a(n) = 13^n - A322089(n).

a(n) = Sum_{i=0..n-1} A322092(i)*13^i.

a(n) = A286840(n)*A322085(n) mod 13^n = A286841(n)*A322086(n) mod 13^n.

EXAMPLE

7^2 = 49 = 4*13 - 3.

124^2 = 15376 = 91*13^2 - 3 = 7*13^3 - 3.

13306^2 = 177049636 = 6199*13^4 - 3.

PROG

(PARI) a(n) = truncate(-sqrt(-3+O(13^n)))

CROSSREFS

Cf. A286840, A286841, A322085, A322086, A322089, A322092.

Sequence in context: A012086 A074487 A192566 * A304420 A217910 A316276

Adjacent sequences:  A322087 A322088 A322089 * A322091 A322092 A322093

KEYWORD

nonn

AUTHOR

Jianing Song, Nov 26 2018

STATUS

approved

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Last modified May 23 18:08 EDT 2022. Contains 353993 sequences. (Running on oeis4.)