FACTORS OF SUMS OF POWERS OF BINOMIAL COEFFICIENTS NEIL J. CALKIN Abstract. We prove divisibility properties for sus of powers of bioial coefficiets ad of -bioial coefficiets. Dedicated to the eory of Paul Erdős It is well ow that if 1. Itroductio f,a = =0 ( ) a the f,0 = + 1, f,1 = 2, f,2 = ( ) 2, ad it is possible to show (Wilf, persoal couicatio, usig techiues i [8]) that for 3 a 9, there is o closed for for f,a as a su of a fixed uber of hypergeoetric ters. Siilarly, usig asyptotic techiues, de Bruij has show [2] that if a 4, the h 2,a has o closed for, where h,a = =0( 1) ( (clearly, h 2+1,a = 0.) I this paper we will prove that while o closed for ay exist, there are iterestig divisibility properties of f,2a ad h 2,a. We will illustrate soe of the techiues which ay be applied to prove these sorts of results. Our ai results are: Theore 1. for all positive ad a, ( ) 2 2 =0( 1) ( 2 Theore 2. For all positive itegers a,, j [ ] a t(, ) ( 1) j, =0 where the [ ] are the -bioial coefficiets, ad t(, ) is a iteger polyoial i with the property that t(, 1) is the odd part of ( ). ) a ) a. 1991 Matheatics Subject Classificatio. 05A10,05A30,11B65. 1
2. Bacgroud I atteptig to exted the results of previous wor [1], we were led to cosider factorizatios of sus of powers of bioial coefficiets. It uicly becae clear that for eve expoets, sall pries occurred as divisors i a regular fashio (Propositio 3), ad that this result could be exteded (Propositio 7) to odd expoets ad alteratig sus. Further ivestigatio revealed (Propositio 8) that for all alteratig sus, the pries dividig h 2,a coicided with those dividig ( ) 2. This led us to cojecture, ad subseuetly to prove, Theore 1: as part of our proof we obtai (Theore 2) a correspodig result for -bioial coefficiets. 3. No-alteratig Sus Propositio 3. For every iteger 1, if p is a prie i the iterval < p < 2a( + 1) 1 2a 1 = + 1 + + 1 (2a 1) the p f,2a. I particular, f,2a is divisible by all pries p for which < p < 2a( + 1) 1 2a 1 = + 1 + 2a 1. The followig lea will eable us to covert iforatio about divisors of f,a which are greater tha ito iforatio about divisors less tha. Lea 4. Let = s s 1... 2 1 0 be the expasio of i base p (ad siilarly for = s s 1... 2 1 0 ). The s f,a f i,a (od p). Proof: By Lucas Theore (see for exaple Graville [6]), ( ) ( ) s i (od p) i=0 i=0 i where as usual, ( ) i i 0 (od p) if i > i. Hece all the ters i the su over for which i > i for soe i disappear, givig ( ) a f,a = as claied. s s 1 s=0 s 1 =0 s i=0 i =0 s i i=0 =0 ( s i 0 0 =0 i=0 ( i i i ) a (od p) f i,a (od p) ) a (od p) Corollary 5. If l < p ad p f l,a the p f l+jp,a for all positive itegers j. 2
We are ow i a positio to prove Propositio 3: we proceed i two stages: first, the case whe < p. Lea 6. Let p be a prie i the iterval < p < 2a(+1) 1 2a 1. The p f,2a. Proof: Let p = + r where r > 0. The we have ( ) 2a p r ( ) 2a p r f,2a = (od p) =0 =0 p r ( ) 2a r + 1 ( 1) 2a (od p) =0 p r ( ) 2a r + 1 (od p) =0 p r ( ) 2a ( + 1)( + 2)... ( + r 1) (od p). (r 1)! =0 Writig x (0) = 1 ad x (r) for the polyoial x(x + 1)... (x + r 1) this last su becoes p r ( ) 2a ( + 1)(r 1). (r 1)! =0 We ow observe that the polyoials x (0), x (1),... x (d) for a iteger basis for the space of all iteger polyoials of degree at ost d. Hece there exist itegers c 0, c 1,... c (r 1)(2a 1) so that Thus f,2a (( + 1) (r 1) ) 2a 1 = p r 1 (r 1)! 2a =0 (r 1)(2a 1) j=0 (r 1)(2a 1) 1 (r 1)! 2a j=0 (r 1)(2a 1) j=0 p r =0 c j ( + r) (j). c j ( + 1) (r 1) ( + r) (j) c j ( + 1) (r+j 1) (r 1)(2a 1) 1 (p r + 1) (r+j) c (r 1)! 2a j. r + j j=0 Now, if r + (r 1)(2a 1) < p, the each of the ters i the su is divisible by p, ad (r 1)! is ot divisible by p: hece f,2a is divisible by p. But ad if ad oly if r + (r 1)(2a 1) = 2ra 2a + 1 = 2pa 2a 2a + 1 copletig the proof of the lea. 2pa 2a 2a + 1 < p p < 2a( + 1) 1 2a 1 3
Now, suppose that = ( 1)p + l with l > 0 ad l < p < 2a(l + 1) 1. 2a 1 The by lea 6, p divides f l,2a ad hece by corollary 5, p divides f,2a. But l < p if ad oly if < p, ad 2a(l + 1) 1 p < 2a 1 if ad oly if 2a( ( 1)p) 1 p < 2a 1 that is if 2a( + 1) 1 p <. 2a 1 Thus, if 2a( + 1) 1 < p < 2a 1 the p divides f,2a, copletig the proof of Propositio 3. 4. Alteratig Sus We ote that o siilar result holds for the case of odd powers of bioial coefficiets (with the trivial exceptio of a=1). Ideed, except for the power of 2 dividig f,2a+1 (which we discuss i Lea 12), the factorizatios of sus of odd powers see to exhibit o structure: for exaple, f 28,3 = 2 6.661.3671.5153.313527009031. However, for alteratig sus of odd powers, we have Propositio 7. p divides h 2,2a+1 for pries i the itervals < p < (2a + 1)( + 1) 1 (2a + 1) 1 = + 1 + + 1 (2a + 1) 1 Proof: ideed by exaiig the proof of Propositio 3, we see that if we defie ( ( )) a g,a = ( 1) =0 so that g,2a = f,2a ad g,2a+1 = h,2a+1, the g,a is divisible by all pries i each of the itervals ( + 1)a 1 < p < a 1 so Propositios 3 ad 7 are really the sae result. For all alteratig sus we have Propositio 8. If p ( 2 ) the p h2,a. 4
Proof: Clearly 2 divides h 2,a if ad oly if 2 divides the iddle ter, ( ) 2 a, as all of the other ters cacel (od 2). Sice 2 divides ( ) 2, 2 divides h2,a. Now let p be a odd prie dividig ( ) 2 : we will show that p divides h2,a. By Kuer s theore, at least oe of the digits of 2 writte i base p is odd (sice if all are eve, the there are o carries i coputig + = 2 i base p). Let the digits of 2 i base p be (2) s, (2) s 1,... (2) 1, (2) 0. The as i Lea 4 ) 2 a =0( 1) ( 2 (2) s i ( ) a ( 1) (2)i i i=0 i =0 i ), at least oe of the digits of 2 (sice p is odd, ( 1) = ( 1) 0+ 1 +... s ). Now, sice p ( 2 i base p is odd: but the the correspodig ter i the product is zero, ad so p h 2,a, copletig the proof of Propositio 8. After coputig soe exaples, it is atural to cojecture (ad the, of course, to prove!) Theore 1. 5. The Mai Theores We will prove Theore 1 by cosiderig bioial coefficets. Defiitios: Let be a positive iteger: throughout we will deote the uber of 1 s i the biary expasio of by l() (so that 2 l() ( ) 2 ): we further defie the followig polyoials i a ideteriate : (the -euivalet of ) (the th cyclotoic polyoial i ), (the -euivalet of!), ad (the -euivalet of ( ) ). Further, defie θ () = 1 1 = 1 + + 2 + + 1 φ () = (1 d ) µ( d ) d! = θ i () i=1 [ ] =!! ( )! r(x, ) = j x ( 1 j ) = (1 )! ad s(x, ) = ( ) 1 2j+1 t(, ) = 2j+1 x s(, ) s(, )s(, )s(, ).... 2 4 8 5
Note that the apparetly ifiite product i the deoiator is i fact fiite, sice s(x, ) = 1 if x < 1. We ow ae soe useful observatios about t(, ). First, so t(, ) = s(, ) = r(, ) r( 2, 2 ) s(, ) s(, ) s(, ) s(, ) s( 2, 2 )2 s( 4, 4 )2 s( 8, 8 )2 s(,... 16 )2 = r(, 2 )2 r(, 4 )2 r(, 8 )2 r(, ) r(, ) 2 r(, ) 4 r(, 2 2 ) r(, 4 2 ) r(, 8 2... r(, 4 2 ) 2 r(, 8 2 ) 2 r(, 16 2 ) 2 r(, ) = r(, 2 )2 r(, ) 2 r(, 4 )2 r(, 2 2 ) r(, 4 2 ) 2 r(, ) 4 r(, 8 )2 r(, 4 2 ) r(, 8 2 ) 2 where agai, the apparetly ifiite product is i fact fiite. Now, sice as 1, we see that r(x, ) r( x 2, )2 r(x, 2 ) r( x 2, 2 ) 2 li t(2, ) = 1 Further, t(2 + 1, ) has a factor ( 1), so { 1 x eve 1 x odd 2 ( 2 ) 2 l. li t(2 + 1, ) = 0. 1 r( 8, ) r( 16, )2 r( 8, 2 ) r( 16, 2 ) 2 I other words, sice 2 l ( ) 2, we ay regard t(2, ) as the -euivalet of the largest odd factor of ( ) 2. Lea 9. t(, ) = φ ()... where the product is over those odd for which is odd. Proof: Clearly, if is eve the φ () does t divide t(, ). Suppose is odd: the φ () divides s(, ) exactly /2 ties, ad hece φ () divides t(, ) /2 /2 /2 /2... 2 4 2 j ties. Now, by cosiderig the biary expasio of, it is iediate that this is 0 if is eve, ad 1 if is odd. 6
Lea 10. Let,, be o-egative itegers: ad write = + = + ( ) = ( ) + ( ) where, are the least o-egative residues of, (od ). The [ ] [ ] ( ) (od φ ()) where [ ] is tae to be 0 if <. Proof: We cosider polyoials odulo θ () ad φ (). First, observe that if ad oly if that is, if ad oly if (od ) 1 1 (od 1 ), θ () θ () θ () θ () (od θ ()), (od φ ()). Now, [ ] r(, ) = r(, )r(, ) ad reducig those ters which are coprie to φ () we obtai r( 1, ) ( ) r(, ) r(, ) r(, )r(( ), ) r(, )r(( ), ). Now, = + ( ) the ( ) = 0, ad [ ] [ ] r(, ) r(, )r(( ), ) (od φ ()) [ ] ( ) (od φ ()) sice θ j () j (od (1 )), ad hece (od φ ()). If + = + ( ), the ( ) = 1, ad [ ] 0 (od φ ()), ad sice >, we have as reuired. [ ] [ ] ( ) (od φ ()) 7
We ote that evaluatig [ ] at = 1 iediately iplies Kuer s theore, sice [ ] is a product of cyclotoic polyoials, ad sice φ (1) = p if = p i ad 1 otherwise, we have a factor of p correspodig to each positio for which there is a carry i = + ( ) base p. We are grateful to Professor Ira Gessel for iforig us that Lea 10 appears without proof as a property of -bioial coefficiets i [7], as Propositio 2.2 i [4] ad as Rear 2.4 i [3]. Proof of Theore 2: It is eough to show that if ad = are odd, the [ ] a φ () ( 1) j. But, fro Lea 10, [ ] a ( 1) j =0 [ = =0 =0 =0 ] a =0 [ ] a ( ) a ( 1) + j (od φ ()) ( 1) j =0 ( ) a ( 1) j ad sice ad are odd, the secod su is zero, ad we are doe. We observe ow that both sides of Theore 2 are iteger polyoials; thus whe we evaluate the at = 1, the left had side (if o-zero) will divide the right had side. But we have already observed that t(2, 1) = ( ) 2 /2 l, ad hece we have proved Corollary 11. ( ) ( ) 2 2 a 2 2 l() ( 1). =0 To prove Theore 1 it reais to show that 2 l() We prove a stroger result by iductio. 2 =0 Lea 12. For all positive itegers a ad, 2 l() ad 2 l() =0 ( ) a 2 ( 1). =0 ( ) a. ( ) a ( 1). Proof: The theore is clearly true whe = 1: assue ow that it it holds for all values less tha. For each 1 i l(), let = 2 i ad let,,,, ( ), ( ) be defied as i Lea 10: writig w i () = [ ] a ( ( ) a ) 8
we have [ ] a w i () (od φ 2 i()) =0 By our iductio hypothesis, sice l() = l( ) + l( ), 2 l() w i (1) for each i. We ow wish to cobie these euivaleces odulo θ 2 l()() = φ 2 ()φ 4 ()φ 8 ()... φ 2 l()() ad evaluate the at = 1. To do this, defie ad for i = 2, 3,..., l(). The, settig π 1 = 1 2 l 1 π i = 1 (1 ) 2l i+1 u i () = φ 2 ()φ 4 ()... φ 2 i 1()π i φ 2 i+1()... φ 2 l()() we have u 1 () = 1 2 (1 + l 1 2 )(1 + 4 )... (1 + 2l() 1 ) 1 (od (1 + )) ad for i 2, u i () 1 2 (1 l i+1 2 )(1 + 2 )(1 + 4 )... (1 + 2i 2 )(1 + 2i )... (1 + 2l() 1 ) Further, if i j, Hece, that is, 1 (1 2i 1 )(1 + 2i )(1 + 2i+1 )... (1 + 2l() 1 ) 2l i+1 1 (od (1 + 2i 1 )) =0 [ ] a u i () 0 (od φ 2 j()). l() i=1 w i ()u i () ( od θ 2 l()()) [ ] a l() = P ()θ =0 2 l()() + w i ()u i () i=1 where we wish to coclude that P () is a iteger polyoial. Observe that it is sufficiet to prove that each w i ()u i () is a iteger polyoial, sice θ 2 l() is oic. To do this, cosider w i (): first, observe that w i () is divisible by 2 l() by our iductive hypothesis, sice < : further, if is odd, so is, ad hece the -bioial su i w i () is syetric ad its coefficiets are eve: if is eve, the l( ) i 1, ad i each case, 2 l i w i () (that is, each coefficiet of w i () is divisible by 2 l i+1 ). Thus, for each i, w i ()u i () is a iteger polyoial. We have thus prove that =0 [ ] a = P ()θ 2 l()() + 9 l() i=1 w i ()u i ()
where P () has iteger coefficiets. Now, settig = 1 i both sides, we observe that u i (1) is a iteger for each i, 2 l() w i (1) for each i (ideed, u i (1) = 0 for i 2, ad u 1 (1) = 1), ad that θ 2 l()(1) = 2 l(). Hece each ter o the right is divisible by 2 l(), provig that ( ) a 2 l(). =0 To prove that ( ) a 2 l() ( 1) =0 we proceed siilarly, settig v i () = [ ] a ( ( ) ( 1) a ), with the oly ajor differece beig i the proof that l() i=1 v i ()u i () is a iteger polyoial: i this case, if is eve, thigs wor as above, ad if is odd, the we have is odd, ad v i () is idetically eual to 0. Note that we eed to have already prove the Lea for o-alteratig sus to prove the alteratig case. This copletes the proof of Lea 12 ad thus of Theore 1. We gratefully acowledge ay iforative discussios with Professors Joatha M. Borwei, Ira Gessel, Adrew J. Graville ad Herbert S. Wilf. Refereces [1] Neil J. Cali, A Curious Bioial Idetity, Discrete Math 131 (1994), 335-337. [2] N. G. DeBruij, Asyptotic Methods i Aalysis, (Dover, New Yor 1981). [3] Ma-Due Choi, George A. Elliott, ad Norio Yui, Gauss polyoials ad the rotatio algebra, Ivetioes Matheaticae 99 (1990), 225 246. [4] J. Désaréie, U aalogue des cogrueces de Kuer pour les -obres d Euler, Europ. J. Cobi. 3 (1982), 19 28. [5] A. J. Graville, Zaphod Beeblebrox s Brai ad the Fify-ith Row of Pascal s Triagle, Aerica Math. Mothly 99 (1992), 318 331. [6] A. J. Graville, The Arithetic Properties of Bioial Coefficiets Proceedigs of the Orgaic Matheatics Worshop (1996) http://www.cec.sfu.ca/orgaics/papers/graville/idex.htl (URL verified October 10, 1996). [7] Gloria Olive, Geeralized Powers, Aerica Math Mothly 72 (1965), 619 627. [8] Maro Petovše, Herbert S. Wilf ad Doro Zeilberger, A=B, (A. K. Peters, Wellesley, Massachusetts, 1996). School Of Matheatics, Georgia Istitute of Techology, Atlata, GA 30332 E-ail address: cali@ath.gatech.edu 10