A341752 Successive approximations up to 2^n for the 2-adic integer 17^(1/4). This is the 3 (mod 4) case.

3, 3, 3, 19, 19, 83, 83, 83, 595, 595, 595, 595, 8787, 8787, 41555, 107091, 107091, 107091, 107091, 107091, 2204243, 6398547, 6398547, 23175763, 56730195, 123839059, 123839059, 123839059, 660709971, 1734451795, 1734451795, 1734451795, 1734451795, 18914320979

Offset

2,1

Comments

a(n) is the unique number k in [1, 2^n] and congruent to 1 mod 4 such that k^4 - 17 is divisible by 2^(n+2).

For odd k, k has a fourth root in the ring of 2-adic integers if and only if k == 1 (mod 16), in which case k has exactly two fourth roots.

Links

Jianing Song, Table of n, a(n) for n = 2..1000

Formula

a(2) = 3; for n >= 3, a(n) = a(n-1) if a(n-1)^4 - 17 is divisible by 2^(n+2), otherwise a(n-1) + 2^(n-1).

a(n) = 2^n - A341751(n).

a(n) = Sum_{i=0..n-1} A341754(i)*2^i.

a(n)^2 == A341538(n) (mod 2^n).

Example

The unique number k in [1, 4] and congruent to 3 modulo 4 such that k^4 - 17 is divisible by 16 is 3, so a(2) = 3.
a(2)^4 - 17 = 64 which is divisible by 32, so a(3) = a(2) = 3.
a(3)^4 - 17 = 64 which is divisible by 64, so a(4) = a(3) = 3.
a(4)^4 - 17 = 64 which is not divisible by 128, so a(5) = a(4) + 2^4 = 19.
a(5)^4 - 17 = 130304 which is ndivisible by 256, so a(6) = a(5) = 19.
...

Prog

(PARI) a(n) = truncate(-sqrtn(17+O(2^(n+2)), 4))

Crossrefs

Cf. A341754 (digits of the associated 2-adic fourth root of 17), A341538.

Approximations of p-adic fourth-power roots:

A341751, this sequence (2-adic, 17^(1/4));

A325484, A325485, A325486, A325487 (5-adic, 6^(1/4));

A324077, A324082, A324083, A324084 (13-adic, 3^(1/4)).

Sequence in context: A342363 A229934 A239125 * A325892 A127014 A073748

Adjacent sequences: A341749 A341750 A341751 * A341753 A341754 A341755

Keyword

nonn

Author

Jianing Song, Feb 18 2021

Status

approved