GRACEFUL NUMBERS. KIRAN R. BHUTANI and ALEXANDER B. LEVIN. Received 14 May 2001

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1 IJMMS 29: PII S htt://immshindawicom Hindawi Publishing Cor GRACEFUL NUMBERS KIRAN R BHUTANI and ALEXANDER B LEVIN Received 4 May 200 We construct a labeled grah Dn that reflects the structure of divisors of a given natural number n We define the concet of graceful numbers in terms of this associated grah and find the general form of such a number As a consequence, we determine which graceful numbers are erfect 2000 Mathematics Subect Classification: B75, 05C78 Introduction In [2], Gallian resented a detailed survey of various tyes of grah labeling, the two best nown being graceful and harmonious Recall that a grah G with q edges is called graceful if one can label its vertices with distinct numbers from the set {0,,,q} and mar the edges with differences of the labels of the end vertices in such a way that the resulting edge labels are distinct A number of interesting results on graceful and graceful-lie labelings are obtained in [, 3, 4] and some other wors In this note, we give a descrition of natural numbers whose associated grah of divisors satisfies certain graceful-lie conditions For any natural number n, we construct a labeled grah Dn that reflects the structure of divisors of n We define the concet of graceful numben terms of this associated grah and find the general form of such a number As a consequence, we determine which graceful numbers are erfect 2 Main results Given a natural number n one can generate a grah Dn that reflects the structure of divisors of n as follows The vertices of the grah reresent all the divisors of the number n, each vertex is labeled by a certain divisor In what follows, we refer to the vertex of the grah Dn with label as the vertex If r and s are two divisors of n and r>s, then there is an edge between the vertices s and f and only if s divides r and the ratio r/s is a rime number As in the theory of graceful grahs, we label such an edge by the difference r s of the labels of its vertices In what follows, the sum of the labels of all edges of the grah Dn is denoted by SDn while SDn denotes the sum of labels of all edges of Dn excet the edges terminating at n Clearly,ifn = r r 2 2 r is the rime factorization of a natural number n, thensdn = SDn i=n n/ i Examle 2 It is easy to see that if n = r, where is a rime number and s any ositive integer, then SDn = r i= i i = r and SDn = r i= i i = r, so that SDn < n The grah Dn is shown in Figure 2

2 496 K R BHUTANI AND A B LEVIN r r r Figure 2 The grah D r The following examle shows that there are numbers n such that SDn > n, as well as numbers that satisfy the condition SDn = n Examle 22 Let n = 24 and m = 2 Then SDn = = 30 >nand SDm = = 2 = m Definition 23 A natural number n is called graceful if SDn = n In order to obtain the descrition of graceful numbers, we first find the value of SDn when n is a roduct of owers of two different rime numbers Examle 24 Let n = r q s where and q are different rime numbers, r, and s In this case the grah Dn is of the form q s q s q s r q s q S r r q s r q s q r q s q r q s q s r r q S q q q r q q q r q r Figure 22 The grah D r q s and SDn = r s i=0 = i q i q + r s i= =0 i q i q = r i=0 i q s + s =0 q r the first sum corresonds to the differences of the consecutive divisors of n when the exonent of q decreases, and the second sum taes care about the differences of consecutive divisors of n when the exonent of decreases Thus, SDn = q s r i + r s q = q s r + + r q s+ q, 2 i=0 =0

3 GRACEFUL NUMBERS 497 so that [ SDn = SDn n n + n n ] 22 q It follows from formulas 2 and22 that a number n = r q s and q are rime, r, and s is graceful if and only if = 2ands =, that is, n = 4q for some odd rime number q Indeed, equality SDn = n can hold only for even numbers n if n is odd, then 2 shows that SDn is even, whence SDn n If n = 2 r q s, where r 2, s 2, then SDn n = 2 r s q i + q s 2 r + 2 r + q s +2 r q s +2 r q s 2 r q s i=0 > 2 r 2 q s + 2 r + q s q s 2 r r s 2 > 0, i=0 q i 23 so that SDn > n Finally, if n = 2 r qr, thensdn n = q 2 r r q + 2 r + q + 2 r q + 2 r 2 r q = q2 r 2, so that SD2 r q = 2 r q if and only if r = 2 Thus, for any two different rime numbers and q, <q, and for any two nonnegative integers r and s, the number r q s is graceful if and only if = 2, r = 2, and s = Now, we generalize formula 2 to the case of arbitrary number n More recisely, we show that if n = r r 2 2 r is a rime decomosition of a ositive integer n,, are different rimes and r,,r are ositive integers, then SDn = i= i, i 24 r + We roceed by induction on n We have seen that the formula is true if n is a ower of a rime number or a roduct of two owers of rimes In order to erform the ste of induction, notice that n SDn = SD + r r r 2 i 2 =0 r i =0 Alying the inductive hyothesis and taing into account that SDn = r 2 i 2 =0 we obtain that SDn = r =2 r i =0 r + + r r =2 r 2 2 r + i=2 r + i 2 2 i +r n SD r 25 r = i = =2 i=0 i + r =2 r + r i=2 i 2, i 2, i, 26 r + r +

4 498 K R BHUTANI AND A B LEVIN = r r + r + i i = i= =2 i, i r + i=2,, i r + 27 so formula 24 is roved Now, formulas 22 and 24 imly that SDn = i= i, i r + i= n ni 28 Formula 28 shows, in articular, that if a number n is odd, then SDn is even it is easily seen that both sums in the right side of the formula are even if n is odd Therefore, every graceful number must be even, that is, n = 2 r q s q sm m 29 for some odd rimes q,,q m m,s i fori =,,m As we have seen, if m =, then the number n is graceful if and only if s = and r = 2, that is, n = 4q We show that if m 2, then SDn > n, so the only graceful numbers are the numbers of the form 4q where q is an odd rime First of all, notice that SD2 r q s 2 r q s for r, s 2 see Examle 24 and SD2q q 2 2q q 2 for any two different rimes q and q 2 alying formula 2 we obtain that SD2q q 2 = q + q q q q 2 q + 6q q 2 +q q 2 +2q +2q 2 = 2q q 2 +3q +q 2 5 > 2q q 2 Therefore, in order to rove that SDn > n for any number n of the form 29 withm 2, it is sufficient to rove that SDn > qm sm SDn/qm sm But the last inequality is a consequence of equality 25 Indeed, n SDn = SDn n = SD +q r sm q sm q m n n +qm sm SD n>q sm qm sm m SD q m s i=0 i =0 n q m s m i m =0 = q sm m SD n q sm m 2 i q i q i m m 20 We arrive at the following result Theorem 25 odd rime A natural number n is graceful if and only if n = 4q where q is an Recall that a ositive integer m is called a erfect numbef it is equal to the sum of all its roer divisors ie, of all divisors of m excet of the number m itself It is nown cf [4, Theorem 50] that every even erfect numbes of the form 2 2, where the number 2 is rime Thus, our theorem imlies the following result Corollary 26 The only erfect graceful numbes 28

5 GRACEFUL NUMBERS 499 References [] R W Frucht, Nearly graceful labelings of grahs, Sci Ser A Math Sci NS 5 992/93, [2] J A Gallian, A dynamic survey of grah labeling, Electron J Combin 5 998, no, Dynamic Survey 6, 43 [3] D Moulton, Graceful labelings of triangular snaes, Ars Combin , 3 3 [4] A Rosa, On certain valuations of the vertices of a grah, Theory of Grahs Internat Symos, Rome, 966, Gordon and Breach, New Yor, 967, Kiran R Bhutani: Deartment of Mathematics, The Catholic University of America, Washington, DC 20064, USA address: bhutani@cuaedu Alexander B Levin: Deartment of Mathematics, The Catholic University of America, Washington, DC 20064, USA address: levin@cuaedu

6 Mathematical Problems in Engineering Secial Issue on Time-Deendent Billiards Call for Paers This subect has been extensively studied in the ast years for one-, two-, and three-dimensional sace Additionally, such dynamical systems can exhibit a very imortant and still unexlained henomenon, called as the Fermi acceleration henomenon Basically, the henomenon of Fermi acceleration FA is a rocess in which a classical article can acquire unbounded energy from collisions with a heavy moving wall This henomenon was originally roosed by Enrico Fermi in 949 as a ossible exlanation of the origin of the large energies of the cosmic articles His original model was then modified and considered under different aroaches and using many versions Moreover, alications of FA have been of a large broad interest in many different fields of science including lasma hysics, astrohysics, atomic hysics, otics, and time-deendent billiard roblems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos both conservative and dissiative chaos We intend to ublish in this secial issue aers reorting research on time-deendent billiards The toic includes both conservative and dissiative dynamics Paers discussing dynamical roerties, statistical and mathematical results, stability investigation of the hase sace structure, the henomenon of Fermi acceleration, conditions for having suression of Fermi acceleration, and comutational and numerical methods for exloring these structures and alications are welcome To be accetable for ublication in the secial issue of Mathematical Problems in Engineering, aers must mae significant, original, and correct contributions to one or more of the toics above mentioned Mathematical aers regarding the toics above are also welcome Authors should follow the Mathematical Problems in Engineering manuscrit format described at htt://www hindawicom/ournals/me/ Prosective authors should submit an electronic coy of their comlete manuscrit through the ournal Manuscrit Tracing System at htt:// mtshindawicom/ according to the following timetable: Guest Editors Edson Denis Leonel, Deartamento de Estatística, Matemática Alicada e Comutação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 55 Bela Vista, Rio Claro, SP, Brazil ; edleonel@rcunesbr Alexander Losutov, Physics Faculty, Moscow State University, Vorob evy Gory, Moscow 9992, Russia; losutov@chaoshysmsuru Manuscrit Due December, 2008 First Round of Reviews March, 2009 Publication Date June, 2009 Hindawi Publishing Cororation htt://wwwhindawicom