A325892 The successive approximations up to 2^n for the 2-adic integer 3^(1/5).

0, 1, 3, 3, 3, 19, 19, 83, 211, 211, 211, 211, 2259, 6355, 14547, 30931, 63699, 129235, 129235, 129235, 129235, 129235, 129235, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 4323539, 2151807187, 6446774483, 6446774483, 6446774483

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..35.

Wikipedia, p-adic number

Formula

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

Example

For n = 2, the unique solution to x^5 == 3 (mod 4) in the range [0, 3] is x = 3, so a(2) = 3.
a(2)^5 - 3 = 240 which is divisible by 8, so a(3) = a(2) = 3;
a(3)^5 - 3 = 240 which is divisible by 16, so a(4) = a(3) = 3;
a(4)^5 - 3 = 240 which is not divisible by 32, so a(5) = a(4) + 16 = 19;
a(5)^5 - 3 = 2476096 which is divisible by 64, so a(6) = a(5) = 19.

Prog

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

Crossrefs

For the digits of 3^(1/5), see A325896.

Approximations of p-adic fifth-power roots:

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

A325893 (2-adic, 5^(1/5));

A325894 (2-adic, 7^(1/5));

A325895 (2-adic, 9^(1/5));

A322157 (5-adic, 7^(1/5));

A309450 (7-adic, 2^(1/5));

A309451 (7-adic, 3^(1/5));

A309452 (7-adic, 4^(1/5));

A309453 (7-adic, 5^(1/5));

A309454 (7-adic, 6^(1/5)).

Sequence in context: A229934 A239125 A341752 * A127014 A377278 A073748

Adjacent sequences: A325889 A325890 A325891 * A325893 A325894 A325895

Keyword

nonn

Author

Jianing Song, Sep 07 2019

Status

approved