Download Parity and Primality of Catalan Numbers

References
1. J.W. Brown and R. V. Churchill, Complex Variables and Applications,
6th ed., McGraw-Hill,
2. E. S. Loomis, The Pythagorean
National Council of Teachers of Mathematics,
Proposition,
3. O. Shisha, On Ptolemy's
theorem, Internat. J. Math. & Math. Sei., 14 (1991), 410.
Parity
and Primality
Thomas
Koshy
([email protected])
State College,
Framingham
mass.edu),
of Catalan
1996.
1968.
Numbers
and Mohammad
MA
Framingham,
Salmassi
01701
(msalmas@frc.
Catalan numbers,
like Fibonacci
and Lucas numbers, appear in a variety of situ
ations, including the enumeration of triangulations of convex polygons, well-formed
sequences of parentheses, binary trees, and the ballot problem [l]-[5]. Like the other
Catalan
families,
for
exploration,
are
numbers
a great
source
and
experimentation,
of
and
pleasure,
are
excellent
candidates
conjecturing.
who
They are named after the Belgian mathematician
Eugene Catalan (1814-1894),
discovered
them in his study of well-formed
of
Leon
sequences
parentheses. However,
hard Euler (1707-1783)
had found them fifty years earlier while counting the number
of triangulations of convex polygons
[3]. But the credit for the earliest known discov
Antu Ming
who was aware
ery goes to the Chinese mathematician
(ca. 1692-1763),
of them as early as 1730 [6].
In 1759 the German mathematician
and physicist
von Segner
Johann Andreas
a
of
that
found
the
number
of
(1707-1777),
Euler,
contemporary
Cn
triangulations of
a convex polygon satisfies the recursive formula
Cn
?
CoCn-i
+
CiCn_2
+
+
Cn_iCo,
(1)
where Co = 1 [3,5]. The numbers Cn are now called Catalan numbers. It follows from
(1) that Ci = 1, C2 = 2, C3 = 5, and so on.
Using generating functions and Segner's formula, an explicit formula for Cn can be
[5]:
developed
(2n)!
(n+iy.nl
"
n+
1\ n J
Cn can be extracted from Pascal's
Consequently,
mial coefficient
in row 2n by n + 1.
(2n")
In
this
note,
we
identify
Catalan
numbers
that
triangle by dividing
are
odd,
Table 1, which gives the first eighteen Catalan numbers,
with asterisks and those that are prime with daggers.
and
those
the central bino
that
are
prime.
In
those that are odd are marked
It follows from the table that when n < 17, Cn is odd
Numbers.
Parity of Catalan
for n = 0, 1, 3, 7, and 15, all of which are of the form 2m ? 1.When m > 0, such
numbers
Theorem,
52
are
known
as Mersenne
numbers.
For n > 0, Cn is odd if and only ifn is aMersenne
number.
ASSOCIATIONOF AMERICA
? THE MATHEMATICAL
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1. The
Table
first
18 Catalan
Numbers.
4
5
6
7
8
5*1"
14
42
132
429*
1430
12
13
15
16
77?0123
Cn
1
1*
21"
n
9
10
11
Cn
4862
16796
=
208012
2674440
742900
from the recurrence
It follows
Proof.
Cn
58786
14
35357670
9694845*
129644790
(1) that
relation
2(C0Cn-i
+ CiC?_2
H-h
C{n/2)-\Cn/2)
2(C0Cn_i
+
H-h
C(?_3)/2C(?+i)/2)
CiC?_2
17
ifn
even
is
+
otherwise.
C2n_l)/2
for n > 0, Cn is odd if and only if both n and C(n-\)/2 are odd. The
Consequently,
same argument implies that Cn is odd if and only if (n ? l)/2 and C(n_3)/4 are both
?
= 0.
this finite descent, it follows that Cn is odd if and
odd or (n
Continuing
l)/2
> 1. But the least value of k for which C* is
m
is
where
if
odd,
only
Cn-(2m-\)/2m
odd is k = 0. Thus the sequence of these if and only if statements terminates when
n
-
-
(2m
l)/2m
=
0;
that
is, when
n =
2m
-
1, a Mersenne
number.
to Table 1, we make another observa
Numbers.
of Catalan
Returning
Primality
are
theorem confirms
two
numbers
of
these
Catalan
tion: Exactly
prime. The following
this.
Theorem.
The only prime Catalan
numbers
are C2 and C3.
It follows from (2) that (n + 2)Cn+i =
Proof.
prime for some n. It follows from (1) that if n >
?
n + 2. Consequently,
kCn
Cn \Cn+\, so Cn+\
1 < & < 3 and thus n
An + 2 = &(rc+ 2), whence
the only Catalan numbers that are prime.
Acknowledgment.
comments
thoughtful
The
and
authors
like
would
suggestions
for
to
improving
thank
that Cn is a
(An + 2)Cn. Assume
3, then (n + 2)/Cn < 1; so Cn >
for some positive
integer k. Then
< 4. It follows that C2 and C3 are
the
referees
the original
and
version
the
of
editor
for
their
the manuscript.
References
1978.
1. D.I. A. Cohen, Basic Techniques of Combinatorial
Theory, Wiley,
to Probability
Vol. 1, 3rd ed., Wiley,
2. W. Feller, An Introduction
Theory and Its Applications,
inUnexpected
3. M. Gardner, Catalan Numbers: An Integer sequence thatMaterializes
Places,
ican, 234 (1976) 120-125.
1968.
Scientific
Amer
63 (1990) 3-20.
4. R. K. Guy, The Second Strong Law of Small Numbers, Mathematics
Magazine,
Elsevier Academic
with Applications,
5. T. Koshy, Discrete Mathematics
Press, 2004.
of the Catalan Numbers, Mathematical
6. P. J. Larcombe, The 18th Century Chinese Discovery
Spectrum,
(1999)5-7.
VOL. 37, NO. 1, JANUARY2006 THE COLLEGE MATHEMATICSJOURNAL 53
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