* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Parity and Primality of Catalan Numbers
Survey
Document related concepts
History of logarithms wikipedia , lookup
Positional notation wikipedia , lookup
Law of large numbers wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Infinitesimal wikipedia , lookup
History of mathematics wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Ethnomathematics wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Non-standard analysis wikipedia , lookup
Real number wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Large numbers wikipedia , lookup
Collatz conjecture wikipedia , lookup
Transcript
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 This content downloaded from 152.3.25.151 on Fri, 20 Jun 2014 12:46:11 PM All use subject to JSTOR Terms and Conditions 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 This content downloaded from 152.3.25.151 on Fri, 20 Jun 2014 12:46:11 PM All use subject to JSTOR Terms and Conditions 32