OFFSET
0,3
COMMENTS
The other approximation for the 5-adic integer sqrt(-4) with numbers 4 (mod 5) is given in A269590.
For the two approximations of the 5-adic integer sqrt(-1) see A048899 and A048898. For comments and some proofs see A210848.
For the digits of this 5-adic integer sqrt(-4), that is the scaled first differences, see A269591. This 5-adic number has the digits read from the right to the left ...32234111132111101130213043113443324032021 = u.
The companion 5-adic number -u has digits ....12210333312333343314231401331001120412424. See A269592.
The recurrence given below (for n >= 1) has been derived from the following facts (i) x^2 + 4 == 0 (mod 5) has the two distinct solutions x(1) = 1 and x(2) = 5 - x(1) = 4. This guarantees the existence of a unique solution x = x1(n) of x^2 + 4 == 0 (mod 5^n) , for n >= 2 , which satisfies also x1(n) == 1 (mod 5). See e.g., Theorem 50, p. 87 of the Nagell reference. The same is true for the solution x = x2(n) with x2(n) == 4 (mod 5^n). (ii) As a consequence of Hensel's lemma (see e.g., the Wikipedia reference under Hensel lifting) one knows that x1(n) (which is treated here) satisfies the congruence x1(n) == x1(n-1) (mod 5^(n-1)) with x1(1) = x1 = 1. (A similar statement holds for x2(n) with input x2(1) = x(2) = 4. This is used in A269590). These two facts allow one to derive a recurrence for x1(n) (and for x2(n)).
REFERENCES
Trygve Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, p. 87.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..1430
Wikipedia, Hensel's lemma.
FORMULA
Recurrence for n >= 1: a(n) = modp( a(n-1) + 2*(a(n-1)^2 + 4), 5^n), n >= 2, with a(1) = 1. Here modp(a, m) is used to pick the representative of the residue class a modulo m from the smallest nonnegative complete residue system {0, 1, ... , m-1}.
a(n) = 5^n - A269590(n), n >= 1.
a(n) == Lucas(5^n) (mod 5^n). - Peter Bala, Nov 10 2022
EXAMPLE
n=2: 11^2 + 4 = 125 == 0 (mod 5^2), and 125 is the only solution from {0, 1, ..., 24} which is congruent to 1 modulo 5.
n=3: the only solution of x1^2 + 4 == 0 (mod 5^3) with x1 from {0, ..., 124} and x1 == 1 (mod 4) is also 11. The number 114 satisfies also the first congruence but not the second one: 114 == 2 (mod 4).
MAPLE
with(padic):D1:=op(3, op([evalp(RootOf(x^2+4), 5, 20)][1])): 0, seq(sum('D1[k]*5^(k-1)', 'k'=1..n), n=1..20);
# alternative program - see A144837
a := proc (n) option remember; if n = 1 then 1 else irem( a(n-1)^5 + 5*a(n-1)^3 + 5*a(n-1), 5^n) end if; end: seq(a(n), n = 1..20); # Peter Bala, Nov 14 2022
PROG
(PARI) a(n) = truncate(sqrt(-4+O(5^(n)))); \\ Michel Marcus, Mar 04 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Mar 02 2016
STATUS
approved