A290803 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-3). These are the numbers congruent to 2 mod 7 (except for the initial 0).

0, 2, 37, 37, 2095, 14100, 47714, 165363, 988906, 29812911, 271934553, 1401835549, 9311142521, 36993716923, 133882727330, 2846775018726, 12341898038612, 145273620317016, 1308426190253051, 1308426190253051, 35505111746372480, 354674176936820484

Offset: 0

Author

Seiichi Manyama, Aug 11 2017

Keywords

• nonn
• easy

Comments

x = ...256052,

x^2 = ...666664 = -3.

Examples

			a(1) =     2_7 = 2,
a(2) =    52_7 = 37,
a(3) =    52_7 = 37,
a(4) =  6052_7 = 2095,
a(5) = 56052_7 = 14100.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1184
• Peter Bala, Using Lucas polynomials to find the p-adic square roots of -1, -2 and -3, Dec 2022.
• Wikipedia, Hensel's Lemma.

Crossrefs

Cf. A114525, A290796, A290804.

Programs

PARI

a(n) = if (n, truncate(sqrt(-3+O(7^(n)))), 0)

Formula

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

a(n) = L(7^n,2) (mod 7^n) = ( (1 + sqrt(2))^(7^n) + (1 - sqrt(2))^(7^n) ) (mod 7^n), where L(n,x) denotes the n-th Lucas polynomial of A114525. - Peter Bala, Nov 28 2022