A269592 Digits of one of the two 5-adic integers sqrt(-4). Here the ones related to A269590.

4, 2, 4, 2, 1, 4, 0, 2, 1, 1, 0, 0, 1, 3, 3, 1, 0, 4, 1, 3, 2, 4, 1, 3, 3, 4, 3, 3, 3, 3, 2, 1, 3, 3, 3, 3, 0, 1, 2, 2, 1, 2, 0, 0, 4, 1, 3, 0, 4, 1, 1, 3, 4, 3, 1, 1, 2, 1, 1, 1, 0, 0, 1, 3, 1, 3

Offset

0,1

Comments

This is the scaled first difference sequence of A269590.

The digits of the other 5-adic integer sqrt(-4), are given in A269591. See also A268922 for the two 5-adic numbers -u and u.

a(n) is the unique solution of the linear congruence 2*A269590(n)*a(n) + A269594(n) == 0 (mod 5), n>=1. Therefore only the values 0, 1, 2, 3 and 4 appear. See the Nagell reference given in A268922, eq. (6) on p. 86, adapted to this case. a(0) = 4 follows from the formula given below.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

BCMATH Congruence Programs, Finding a p-adic square root of a quadratic residue (mod p), p an odd prime..

Formula

a(n) = (b(n+1) - b(n))/5^n, n>=0, with b(n) = A269590(n), n >= 0.

a(n) = -A269594(n)*(2*A269590(n))^3 (mod 5), n >= 1. Solution of the linear congruence see, e.g., Nagell, Theorem 38 pp. 77-78.

A269590(n+1) = sum(a(k)*5^k, k=0..n), n>=0.

a(n) = 4 - A269591(n) if n > 0 and a(0) = 5 - A269591(0) = 5-4 = 1. - Michel Marcus, Mar 31 2016

Example

a(4) = -212*(2*364)^3 (mod 5) = -2*(2*(-1))^3 (mod 5) = 1.

Prog

(PARI) a(n) = truncate(-sqrt(-4+O(5^(n+1))))\5^n; \\ Michel Marcus, Mar 04 2016

Crossrefs

Cf. A269590, A269591 (companion), A269594.

Sequence in context: A179950 A016514 A224360 * A105256 A064127 A178253

Adjacent sequences: A269589 A269590 A269591 * A269593 A269594 A269595

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 02 2016

Status

approved