A322926 The successive approximations up to 2^n for 2-adic integer 5^(1/3).

0, 1, 1, 5, 13, 29, 29, 93, 93, 93, 605, 1629, 3677, 3677, 3677, 20061, 20061, 20061, 151133, 151133, 151133, 151133, 151133, 4345437, 4345437, 21122653, 54677085, 54677085, 188894813, 457330269, 457330269, 457330269, 2604813917, 6899781213, 6899781213

Offset

0,4

Comments

a(n) is the unique solution to x^3 == 5 (mod 2^n) in the range [0, 2^n - 1].

Links

Table of n, a(n) for n=0..34.

Wikipedia, p-adic number

Formula

For n > 0, a(n) = a(n-1) if a(n-1)^3 - 5 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).

Example

13^3 = 2197 = 137*2^4 + 5;
29^3 = 24389 = 762*2^5 + 5 = 381*2^6 + 5;
93^3 = 804357 = 6284*2^7 + 5 = 3142*2^8 + 5 = 1571*2^9 + 5.

Prog

(PARI) a(n) = lift(sqrtn(5+O(2^n), 3))

Crossrefs

For the digits of 5^(1/3), see A323045.

Approximations of p-adic cubic roots:

A322701 (2-adic, 3^(1/3));

this sequence (2-adic, 5^(1/3));

A322934 (2-adic, 7^(1/3));

A322999 (2-adic, 9^(1/3));

A290567 (5-adic, 2^(1/3));

A290568 (5-adic, 3^(1/3));

A309444 (5-adic, 4^(1/3));

A319097, A319098, A319199 (7-adic, 6^(1/3));

A320914, A320915, A321105 (13-adic, 5^(1/3)).

Sequence in context: A226616 A226618 A321770 * A178854 A224339 A368546

Adjacent sequences: A322923 A322924 A322925 * A322927 A322928 A322929

Keyword

nonn

Author

Jianing Song, Aug 30 2019

Status

approved