A290567 The successive approximations up to 5^n for 5-adic integer 2^(1/3).

0, 3, 3, 53, 303, 2178, 5303, 67803, 67803, 849053, 6708428, 6708428, 6708428, 983270928, 983270928, 25397333428, 147467645928, 605231317803, 1368170770928, 5182868036553, 43329840692803, 43329840692803, 43329840692803, 4811701422724053, 52495417243036553

Offset: 0

Author

Seiichi Manyama, Aug 06 2017

Keywords

• nonn

Comments

x = ...132203,

x^2 = ...344214,

x^3 = ...000002 = 2.

Examples

			a(1) = (   3)_5 = 3,
a(2) = (   3)_5 = 3,
a(3) = ( 203)_5 = 53,
a(4) = (2203)_5 = 303.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1431
• Wikipedia, Hensel's Lemma.

Crossrefs

Cf. A290566, A290568.

Programs

PARI

a(n)=truncate((2+O(5^n))^(1/3)); \\ Joerg Arndt, Aug 06 2017

Formula

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