A290804 One of the two successive approximations up to 7^n for the 7-adic integer sqrt(-3). These are the numbers congruent to 5 mod 7 (except for the initial 0).

0, 5, 12, 306, 306, 2707, 69935, 658180, 4775895, 10540696, 10540696, 575491194, 4530144680, 59895293484, 544340345519, 1900786491217, 20891032530989, 87356893670191, 319987407657398, 10090468995120092, 44287154551239521, 203871687146463523

Offset

0,2

Comments

x = ...410615,

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1183

Peter Bala, Using Lucas polynomials to find the p -adic square roots of -1, -2 and -3, Dec 2022.

Wikipedia, Hensel's Lemma.

Formula

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

If n > 0, a(n) = 7^n - A290803(n).

a(n) = L(7^n,5) (mod 7^n) = ( ((5 + sqrt(29))/2)^(7^n) + ((5 - sqrt(29))/2)^(7^n) ) (mod 7^n), where L(n,x) denotes the n-th Lucas polynomial of A114525. - Peter Bala, Nov 28 2022

Example

a(1) =     5_7 = 5,
a(2) =    15_7 = 12,
a(3) =   615_7 = 306,
a(4) =   615_7 = 306,
a(5) = 10615_7 = 2707.

Prog

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

Crossrefs

Cf. A114525, A290797, A290803.

Sequence in context: A195538 A330218 A047658 * A353365 A146542 A264873

Adjacent sequences: A290801 A290802 A290803 * A290805 A290806 A290807

Keyword

nonn,easy

Author

Seiichi Manyama, Aug 11 2017

Status

approved