A269590 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-4). These are the 4 mod 5 numbers (except for n=0).

0, 4, 14, 114, 364, 989, 13489, 13489, 169739, 560364, 2513489, 2513489, 2513489, 246654114, 3908763489, 22219310364, 52736888489, 52736888489, 3104494700989, 6919191966614

Offset: 0

Author

Wolfdieter Lang, Mar 02 2016

Keywords

• nonn
• easy

Comments

The other approximation for the 5-adic integer sqrt(-4) with numbers 1 (mod 5) is given in A268922. See this also for more details and references.

For the digits of this 5-adic integer sqrt(-4), that is the scaled first differences, see A269592. See also A268922 for the 5-adic numbers u and -u written from the right to the left.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1432

Crossrefs

Cf. A048899, A048898, A269591, A268922, A269592.

Programs

Maple

with(padic): D2:=op(3,op([evalp(RootOf(x^2+4),5,20)][2])):
0,seq(sum('D2[k]*5^(k-1)','k'=1..n),n=1..20);
# alternative program
a := proc(n) option remember; if n = 1 then 4 else irem( a(n-1)^5 + 5*a(n-1)^3 + 5*a(n-1), 5^n) end if; end: seq(a(n), n = 1..20); # Peter Bala, Nov 14 2022

PARI

a(n) = if (n==0, 0, 5^n - truncate(sqrt(-4+O(5^(n))))); \\ Michel Marcus, Mar 07 2016

Formula

Recurrence for n >= 1: a(n) = modp( a(n-1) + 3*(a(n-1)^2 + 4), 5^n), n >= 2, with a(1) = 4. Here modp(a, m) is used to pick the representative of the residue class a modulo m from the smallest nonnegative complete residue system {0, 1, ... , m-1}.

a(n) = 5^n - A268922(n), n >= 1.

a(n) == Lucas(3*(5^n)) (mod 5^n). - Peter Bala, Nov 14 2022