A327303 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-9). This is the 4 (mod 5) case (except for n = 0).

0, 4, 4, 79, 79, 79, 3204, 18829, 331329, 1112579, 1112579, 20643829, 118300079, 850721954, 3292128204, 27706190704, 149776503204, 302364393829, 1065303846954, 8694698378204, 46841671034454, 332943965956329, 332943965956329, 5101315547987579, 28943173458143829

Offset: 0

Author

Jianing Song, Sep 16 2019

Keywords

• nonn

Comments

a(n) is the unique number k in [1, 5^n] and congruent to 4 mod 5 such that k^2 + 9 is divisible by 5^n.

Examples

			The unique number k in {4, 9, 14, 19, 24} such that k^2 + 9 is divisible by 25 is k = 4, so a(2) = 4.
The unique number k in {4, 29, 54, 79, 104} such that k^2 + 9 is divisible by 125 is k = 79, so a(3) = 46.
The unique number k in {79, 204, 329, 454, 579} such that k^2 + 9 is divisible by 625 is k = 79, so a(4) = 79.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1424
• G. P. Michon, Introduction to p-adic integers, Numericana.

Crossrefs

For the digits of sqrt(-9) see A327304 and A327305.

Approximations of 5-adic square roots:

A327302, this sequence (sqrt(-9));

A324027, A324028 (sqrt(-6));

A268922, A269590 (sqrt(-4));

A048898, A048899 (sqrt(-1));

A324023, A324024 (sqrt(6)).

Programs

Maple

R:= [padic:-rootp(x^2+9,5,101)]:
R:= op(select(t -> padic:-ratvaluep(t,1)=4, R)):
seq(padic:-ratvaluep(R,n),n=0..100); # Robert Israel, Jan 16 2023

PARI

a(n) = truncate(-sqrt(-9+O(5^n)))

Formula

a(1) = 4; for n >= 2, a(n) is the unique number k in {a(n-1) + m*5^(n-1) : m = 0, 1, 2, 3, 4} such that k^2 + 9 is divisible by 5^n.

For n > 0, a(n) = 5^n - A327302(n).