A325893 The successive approximations up to 2^n for 2-adic integer 5^(1/5).

0, 1, 1, 5, 5, 21, 21, 21, 149, 149, 149, 1173, 1173, 1173, 9365, 9365, 42133, 107669, 238741, 500885, 1025173, 1025173, 1025173, 1025173, 1025173, 1025173, 34579605, 34579605, 34579605, 34579605, 571450517, 1645192341, 1645192341, 1645192341, 10235126933, 10235126933

Offset

0,4

Comments

a(n) is the unique solution to x^5 == 5 (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 - 5 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).

Example

For n = 2, the unique solution to x^5 == 5 (mod 4) in the range [0, 3] is x = 1, so a(2) = 1.
a(2)^5 - 5 = -4 which is not divisible by 8, so a(3) = a(2) + 4 = 5;
a(3)^5 - 5 = 3120 which is divisible by 16, so a(4) = a(3) = 5;
a(4)^5 - 5 = 3120 which is not divisible by 32, so a(5) = a(4) + 16 = 21;
a(5)^5 - 5 = 4084096 which is divisible by 64, so a(6) = a(5) = 21.

Prog

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

Crossrefs

For the digits of 5^(1/5), see A325897.

Approximations of p-adic fifth-power roots:

A325892 (2-adic, 3^(1/5));

this sequence (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: A283212 A283250 A171219 * A146043 A116400 A279810

Adjacent sequences: A325890 A325891 A325892 * A325894 A325895 A325896

Keyword

nonn

Author

Jianing Song, Sep 07 2019

Status

approved