login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A048898 One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-1). Here the 2 (mod 5) numbers (except for n=0). 17

%I

%S 0,2,7,57,182,2057,14557,45807,280182,280182,6139557,25670807,

%T 123327057,123327057,5006139557,11109655182,102662389557,407838170807,

%U 3459595983307,3459595983307,79753541295807,365855836217682,2273204469030182,2273204469030182,49956920289342682

%N One of the two successive approximations up to 5^n for the 5-adic integer sqrt(-1). Here the 2 (mod 5) numbers (except for n=0).

%C This is the root congruent to 2 mod 5.

%C Or, residues modulo 5^n of a 5-adic solution of x^2+1=0.

%C The radix-5 expansion of a(n) is obtained from the n rightmost digits in the expansion of the following pentadic integer:

%C ...422331102414131141421404340423140223032431212 = u

%C The residues modulo 5^n of the other 5-adic solution of x^2+1=0 are given by A048899 which corresponds to the pentadic integer -u:

%C ...022113342030313303023040104021304221412013233 = -u

%C The digits of u and -u are given in A210850 and A210851, respectively. - From _Wolfdieter Lang_, May 02 2012.

%C For approximations for p-adic square roots see also the W. Lang link under A268922. - From _Wolfdieter Lang_, Apr 03 2016.

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

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

%H Vincenzo Librandi, <a href="/A048898/b048898.txt">Table of n, a(n) for n = 0..200</a>

%H G. P. Michon, <a href="http://www.numericana.com/answer/pseudo.htm#witness">On the witnesses of a composite integer</a>, Numericana.

%H G. P. Michon, <a href="http://www.numericana.com/answer/p-adic.htm#integers">Introduction to p-adic integers</a>, Numericana.

%F If n>0, a(n) = 5^n - A048899(n).

%F From _Wolfdieter Lang_, Apr 28 2012: (Start)

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

%F a(n) == 2^(5^(n-1)) (mod 5^n), n>=1.

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

%F (a(n)^2 + 1)/5^n = A210848(n), n>=0.

%F (End)

%F Another recurrence: a(n) = modp(a(n-1) + a(n-1)^2 + 1, 5^n), n >= 2, a(1) = 2. 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

%e a(0)=0 because 0 satisfies any equation in integers modulo 1.

%e a(1)=2 because 2 is one solution of x^2+1=0 modulo 5. (The other solution is 3, which gives rise to A048899.)

%e a(2)=7 because the equation (5y+a(1))^2+1=0 modulo 25 means that y is 1 modulo 5.

%t a[0] = 0; a[1] = 2; a[n_] := a[n] = Mod[a[n-1]^5, 5^n]; Table[a[n], {n, 0, 21}] (* _Jean-Fran├žois Alcover_, Nov 24 2011, after Pari *)

%t Join[{0}, RecurrenceTable[{a[1] == 2, a[n] == Mod[a[n-1]^5, 5^n]}, a, {n, 25}]] (* _Vincenzo Librandi_, Feb 29 2016 *)

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

%o (MAGMA) [n le 2 select 2*(n-1) else Self(n-1)^5 mod 5^(n-1): n in [1..30]]; // _Vincenzo Librandi_, Feb 29 2016

%Y Cf. A000351 (powers of 5), A048899 (the other pentadic number whose square is -1), A034939(n)=Min(a(n), A048899(n)).

%Y Cf. A034935. Different from A034935.

%K nonn,easy,nice

%O 0,2

%A _Michael Somos_, Jul 26 1999

%E Additional comments from _Gerard P. Michon_, Jul 15 2009

%E Edited by _N. J. A. Sloane_, Jul 25 2009

%E Name clarified by _Wolfdieter Lang_, Feb 19 2016

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 10 19:11 EST 2016. Contains 279005 sequences.