A290806 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-5). These are the numbers congruent to 3 mod 7 (except for the initial 0).

0, 3, 17, 311, 997, 3398, 20205, 608450, 2255536, 25314740, 25314740, 307789989, 8217096961, 77423532966, 368090564187, 4437429001281, 4437429001281, 4437429001281, 4437429001281, 3261264624822179, 3261264624822179, 3261264624822179, 1120352992791390193

Offset: 0

Author

Seiichi Manyama, Aug 11 2017

Keywords

• nonn

Comments

x = ...112623,

x^2 = ...666662 = -5.

Examples

			a(1) =    3_7 = 3,
a(2) =   23_7 = 17,
a(3) =  623_7 = 311,
a(4) = 2623_7 = 997.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1182
• Wikipedia, Hensel's Lemma.

Crossrefs

Cf. A290798, A290809.

Programs

Maple

with(padic):
R:= [rootp(x^2+5, 7, 100)]:
R1:= op(select(t -> ratvaluep(evalp(t, 7, 1))=3, R)):
seq(ratvaluep(evalp(R1, 7, n)), n=0..100); # Robert Israel, Aug 13 2017

PARI

a(n) = if (n, truncate(sqrt(-5+O(7^(n)))), 0)

Formula

a(0) = 0 and a(1) = 3, a(n) = a(n-1) + (a(n-1)^2 + 5) mod 7^n for n > 1.