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A048899 One of the two successive approximations up to 5^n for 5-adic integer sqrt(-1). Here the 3 (mod 5) case (except for n=0). 18
0, 3, 18, 68, 443, 1068, 1068, 32318, 110443, 1672943, 3626068, 23157318, 120813568, 1097376068, 1097376068, 19407922943, 49925501068, 355101282318, 355101282318, 15613890344818, 15613890344818, 110981321985443 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is the root congruent to 3 (mod 5) for n>0.

The other case with the 2 (mod 5) numbers (except for n=0) is given in A048898. - Wolfdieter Lang, Feb 19 2016

REFERENCES

J. H. Conway, The Sensual Quadratic Form, p. 118, Mathematical Association of America, 1997, The Carus Mathematical Monographs, Number 26.

K. Mahler, Introduction to p-Adic Numbers and Their Functions, Cambridge, 1973, p. 35.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1431

FORMULA

a(n) = 5^n - A048898(n), n>=1.

a(n) = A066601(5^n), n>=0.

0 <= a(n) < 5^n. 5^n divides a(n)^2 + 1.

From Wolfdieter Lang, Apr 28 2012: (Start)

Recurrence: a(n) = a(n-1)^5 (mod 5^n), a(1) = 3, n>=2. See the Pari program below, and the J.- F. Alcover Mathematica program for A048898.

a(n) = 3^(5^(n-1)) (mod 5^n), n>=1. Compare with the above given formula involving A066601.

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

(a(n)^2 + 1)/5^n = A210849(n), n>=0.

(End)

Another recurrence: a(n) = modp(a(n-1) + 4*(a(n-1)^2 + 1), 5^n), n >= 2, a(1) = 3. Here modp(a, m) is the representative from {0, 1, ... ,|m|-1} of the residue class a modulo m. Note that a(n) is in the residue class of a(n-1) modulo 5^(n-1) (see Hensel lifting). - Wolfdieter Lang, Feb 28 2016

EXAMPLE

a(2) = 18 because the two roots of x^2 + 1 == 0 (mod 5^2) are 7 and 18 and 18 == 3 (mod 5). For 7 see A048898(2).

MATHEMATICA

Join[{0}, RecurrenceTable[{a[1] == 3, a[n] == Mod[a[n-1]^5, 5^n]}, a, {n, 25}]] (* Vincenzo Librandi, Feb 29 2016 * )

PROG

(PARI) {a(n) = if( n<2, 3, a(n - 1)^5) % 5^n}

(MAGMA) [n le 2 select 3*(n-1) else Self(n-1)^5 mod 5^(n-1): n in [1..30]]; // Vincenzo Librandi, Feb 29 2016

CROSSREFS

Cf. A048898, A066601, A210851.

Sequence in context: A110689 A027333 A026576 * A107583 A157535 A098522

Adjacent sequences:  A048896 A048897 A048898 * A048900 A048901 A048902

KEYWORD

nonn,easy

AUTHOR

Michael Somos, Jul 26 1999

EXTENSIONS

Example corrected by Wolfdieter Lang, Apr 28 2012

Name clarified by Wolfdieter Lang, Feb 19 2016

STATUS

approved

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Last modified September 26 01:25 EDT 2017. Contains 292500 sequences.