A325894 The successive approximations up to 2^n for the 2-adic integer 7^(1/5).

0, 1, 3, 7, 7, 7, 39, 103, 231, 231, 743, 1767, 1767, 1767, 1767, 18151, 18151, 18151, 18151, 18151, 542439, 1591015, 3688167, 7882471, 16271079, 33048295, 66602727, 133711591, 267929319, 267929319, 804800231, 804800231, 804800231, 804800231, 9394734823, 26574604007

Offset

0,3

Comments

a(n) is the unique solution to x^5 == 7 (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 - 7 is divisible by 2^n, otherwise a(n-1) + 2^(n-1).

Example

For n = 2, the unique solution to x^5 == 7 (mod 4) in the range [0, 3] is x = 3, so a(2) = 3.
a(2)^5 - 7 = 236 which is not divisible by 8, so a(3) = a(2) + 4 = 7;
a(3)^5 - 7 = 16800 which is divisible by 16, so a(4) = a(3) = 7;
a(4)^5 - 7 = 16800 which is divisible by 32, so a(5) = a(4) = 7;
a(5)^5 - 7 = 16800 which is not divisible by 64, so a(6) = a(5) + 32 = 39.

Prog

(PARI) a(n) = lift(sqrtn(7+O(2^n), 5))

Crossrefs

For the digits of 7^(1/5), see A325898.

Approximations of p-adic fifth-power roots:

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

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

this sequence (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: A123879 A193016 A349604 * A370517 A176269 A343644

Adjacent sequences: A325891 A325892 A325893 * A325895 A325896 A325897

Keyword

nonn

Author

Jianing Song, Sep 07 2019

Status

approved