In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, a juggler sequence is an
integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.
An integer sequence may be specified ''explicitly'' by giving a formula for its ''n''th term, or ''implicitly'' by giving a relationship between its terms. For ...
that starts with a
positive integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
''a''
0, with each subsequent term in the sequence defined by the
recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
:
Background
Juggler sequences were publicised by American mathematician and author
Clifford A. Pickover
Clifford Alan Pickover (born August 15, 1957) is an American author, editor, and columnist in the fields of science, mathematics, science fiction, innovation, and creativity. For many years, he was employed at the IBM Thomas J. Watson Research ...
. The name is derived from the rising and falling nature of the sequences, like balls in the hands of a
juggler
Juggling is a physical skill, performed by a juggler, involving the manipulation of objects for recreation, entertainment, art or sport. The most recognizable form of juggling is toss juggling. Juggling can be the manipulation of one object ...
.
For example, the juggler sequence starting with ''a''
0 = 3 is
:
:
:
:
:
:
If a juggler sequence reaches 1, then all subsequent terms are equal to 1. It is conjectured that all juggler sequences eventually reach 1. This conjecture has been verified for initial terms up to 10
6, but has not been proved. Juggler sequences therefore present a problem that is similar to the
Collatz conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integ ...
, about which
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...
stated that "mathematics is not yet ready for such problems".
For a given initial term ''n'', one defines ''l''(''n'') to be the number of steps which the juggler sequence starting at ''n'' takes to first reach 1, and ''h''(''n'') to be the maximum value in the juggler sequence starting at ''n''. For small values of ''n'' we have:
:
Juggler sequences can reach very large values before descending to 1. For example, the juggler sequence starting at ''a''
0 = 37 reaches a maximum value of 24906114455136. Harry J. Smith has determined that the juggler sequence starting at ''a''
0 = 48443 reaches a maximum value at ''a''
60 with 972,463 digits, before reaching 1 at ''a''
157.
Letter from Harry J. Smith to Clifford A. Pickover, 27 June 1992
/ref>
See also
* Arithmetic dynamics Arithmetic dynamics is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is ...
* Collatz conjecture
The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform every positive integer into 1. It concerns sequences of integ ...
* Recurrence relation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
References
External links
*{{Mathworld, id=JugglerSequence
*Juggler sequence
In number theory, a juggler sequence is an integer sequence that starts with a positive integer ''a''0, with each subsequent term in the sequence defined by the recurrence relation:
a_= \begin
\left \lfloor a_k^ \right \rfloor, & \text a_k \text ...
(A094683) at the On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to the ...
. See also:
** Number of steps needed for juggler sequence (A094683) started at n to reach 1.
** n sets a new record for number of iterations to reach 1 in the juggler sequence problem.
** Number of steps where the Juggler sequence reaches a new record.
** Smallest number which requires n iterations to reach 1 in the juggler sequence problem.
** Starting values that produce a larger juggler number than smaller starting values.
Juggler sequence calculator
at Collatz Conjecture Calculation Center
by Harry J. Smith
Arithmetic dynamics
Integer sequences
Recurrence relations
Unsolved problems in number theory