Divisors of the Middle Binomial Coefficient

Transcription

1 Divisors of the Middle Biomial Coefficiet Carl Pomerace Abstract We study some old ad ew problems ivolvig divisors of the middle biomial coefficiet ( 1 INTRODUCTION I the ceter of the th row of Pascal s triagle we meet the maximal etry ( These middle biomial coefficiets have a rich history For example, the fact that ( is divisible by the product of the primes i the iterval (, was exploited by Chebyshev i 1850 to obtai upper bouds ad lower bouds for the distributio of primes These bouds were so good that it seemed a promisig path to the prime umber theorem, but evetually that goal was reached by other methods The cetral biomial coefficiet ( also figures promietly i the defiitio of the Catala umbers: C( = 1 ( (1 + 1 They too have a rich history with may combiatorial applicatios (see Staley [11] Note that C( is a iteger, that is, + 1 divides ( This paper origiated with the aive questio: Is there some umber other tha 1 such that + always divides (? It turs out that the short aswer is o For each 1 there are ifiitely may with + ot dividig ( ( However,if 2, + usually divides (iaway ( that we will mae precise O the other had, we show that if 0, + divides less frequetly tha ot, leavig uresolved the precise ature of this frequecy While ot particularly deep, these results appear to be ew The proofs stem from a umber-theoretic (as opposed to combiatorial proof that C( is itegral Alog the way we shall meet the otorious problem of Ro Graham (which has a cash prize attached o whether there are ifiitely may umbers with ( relatively prime to 105 But let us begi at the very begiig, taig othig for grated 2 WHY ARE THE BINOMIAL COEFFICIENTS INTEGERS? The biomial coefficiet ( m, defied as m!!(m!, is itegral Really? Perhaps it is ot so obvious that the deomiator divides the umerator There are umerous proofs of course, for example oe ca use iductio ad Pascal s rule: ( ( ( m + 1 m m = MSC: Primary 05A10, Secodary 11B c THE MATHEMATICAL ASSOCIATION OF AMERICA [Mothly 122

2 Or oe ca argue combiatorially that ( m couts the umber of -elemet subsets of a m-elemet set Istead of looig at the shortest proof, let s istead cosider a more complicated proof, oe that allows us to itroduce some useful otatio ad also prove other results Foraprimep ad a positive iteger m, let v p (m deote the umber of factors of p i the prime factorizatio of m For example, v 2 (10 = 1, v 2 (11 = 0, ad v 2 (12 = 2 This fuctio ca be exteded to positive ratioal umbers via v p (a/b = v p (a v p (b Thus, a/b is itegral if ad oly if v p (a/b 0 for all primes p The power of p i m! is give by v p (m! = m v p ( = =1 m =1 v p ( j=1 1 = j 1 m 1 = m j 1 =1, the last step usig that m/ is the umber of multiples of i {1, 2,,m} This result, sometimes referred to as Legedre s formula, is well ow i elemetary umber theory We also have the almost trivial iequality x + y x + y (2 for all real umbers x, y From these two results, it is immediate that ( m is a iteger Ideed, for each prime p we have (( m v p = ( m j 1 m 0 = 0 (3 j 1 3 KUMMER S THEOREM AND CATALAN NUMBERS There is aother cosequece of this lie of thiig For a real umber x let {x} =x x, the fractioal part of x The iequality (2 ca be improved to a equatio: x + y x y ={x}+{y} {x + y} = { 1, if {x}+{y} 1, 0, if {x} +{y} < 1 So by (3, v p ( ( m is the umber of values of j such that {/ }+{(m / } is at least 1 Let s write ad m i the base p, so that = a 0 + a 1 p +, m = b 0 + b 1 p +, where the digits a i, b i are itegers i the rage 0 to p 1 For j 1, { } = a 0 + a 1 p + +a j 1 1, { m } = b 0 + b 1 p + +b j 1 1, ad so we see that {/ }+{(m / } 1 if ad oly if i the additio of ad m i the base p there is a carry ito place j caused by the earlier digits This implies the remarable result of Kummer from 1852: For each prime p ad itegers 0 m, v p (( m is the umber of carries i the additio + (m = m whe doe i the base p August September 2015] MIDDLE BINOMIAL COEFFICIENT 637

3 Recall the defiitio (1 of the Catala umber C( Ay of the umerous combiatorial applicatios of the Catala umbers lead to a proof that C( is a iteger, beig the solutio to a coutig problem From a umber-theoretic perspective, oe ca as for a direct proof of the itegrality of C( This is ot difficult, perhaps the easiest way to see it is via the idetity C( = ( ( 1 It also follows from Kummer s theorem Ideed, say p is a prime ad v p ( + 1 = j The the least sigificat j digits i base p of + 1 are all 0, so the least sigificat j digits i base p of are all p 1 Thus, i the additio + = performed i the base p, wehave j carries from the least sigificat j digits, ad perhaps some other carries as well So we have v p ( ( j Sice this is true for all primes p, wehave + 1 ( ( ad so C( = /( + 1 is a iteger It may seem that the path of this paper is to give difficult proofs of easy theorems! However, we shall see that this proof of the itegrality of C( has legs ad ca be used to also prove some perhaps surprisig ew results But first we tae a oturelated detour to view the otorious 105 problem 4 WHEN IS ( RELATIVELY PRIME TO 105? The umbers = 1, 10, ad 756 have ( relatively prime to 105 Are there ifiitely may others? This problem is due to Ro Graham, ad accordig to [2, 4], Graham offers a prize of $1,000 to settle it What s the deal with 105? It is 3 5 7, the product of the first three odd primes More geerally, oe ca as for ay fixed umber m divisible by at least three distict odd primes, if ( is relatively prime to m for ifiitely may I the case whe m = pq, the product of just two odd primes, we do ow that ( is relatively prime to m for ifiitely may, a result of Erdős, Graham, Ruzsa, ad Straus, see [3] These problems ad results stem from the poit of view tae i Kummer s theorem, discussed i the previous sectio We ca use Kummer s theorem to show that ( is usually divisible by all small primes p This is obvious for p = 2 sice i the base 2, the umber has oe more digit tha so there is at least oe carry i the additio + ; that is, ( is always eve For a odd prime p, let R p = {0, 1,, 12 (p 1 }, r p = #R p = 1 2 (p + 1, θ p = log r p log p We see that p does ot divide ( precisely whe all of the base-p digits of come from R p We show this is a uusual evet Lemma 1 For each odd prime p ad all real umbers x 2, the umber of itegers 1 x with p ( is at most px θ p Proof If p (, the by Kummer s theorem, every base-p digit of is i R p Let D = 1 + log x/ log p, so that if 1 x is a iteger, the has at most D base-p digits If we restrict these digits so that they are i R p, we would have at most 638 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Mothly 122

4 r p choices i each place Thus, the umber of choices for is at most r p D It remais to ote that so cocludig the proof r D p log x/ log p < pr p = px θ p, For example, there are at most 3x θ 3 itegers x with ( ot divisible by 3 Sice the expoet θ 3 = log 2/ log 3 = is smaller tha 1, we see that the vast majority of itegers up to x have ( divisible by 3 Here is a heuristic argumet for why there are ifiitely may with ( relatively prime to 105 The idea is to view Lemma 1 as a assertio about the probability that p does ot divide ( whe is radomly chose i [1, x] Whe x is a iteger, this probability is at most px θp 1 The expoets θ p 1forp = 3, 5, 7 are greater tha 037, 032, ad 029, respectively If these evets are idepedet, as would seem oly just (why would the base-p expasio of have aythig to do with the base-q expasio whe p ad q are differet primes?, the the probability that ( is relatively prime to 105, where x, exceeds x 098 whe x is large Thus, we expect at least x 002 examples, ad this expressio teds to ifiity whe x, albeit fairly slowly It is iterestig to ote that if oe redoes this heuristic for the four primes 3, 5, 7, 11, the ( it suggests that there are at most fiitely may umbers, such as = 3160, where is relatively prime to 1155 No example larger tha 3160 is ow, though they have bee searched for up to , see [7] 5 HOW FREQUENTLY DOES + DIVIDE (? We have see that + 1 divides ( for all We ow as what happes with + whe 1 Is there a value of where + divides ( for all or for all sufficietly large? We prove the followig results which show a importat cleavage betwee the cases whe 2 ad the cases whe 0 However, we first state a uiversal result Theorem 1 For each iteger 1 there are ifiitely may positive itegers with + ( Thus, the case = 1 of Catala umbers is ideed special However, the set of umbers satisfyig the coditio of Theorem 1 whe 2 is rather sparse To measure how dese or sparse a set S of positive itegers is, let S(x deote the umber of members of S i [1, x] The the asymptotic desity of S is lim x S(x/x if this limit exists I geeral, the limsup gives the upper asymptotic desity of S ad the limif the lower asymptotic desity For example, the set of odd umbers has asymptotic desity 1, the set of prime umbers has asymptotic desity 0, ad the set of umbers 2 which have a eve umber of decimal digits has upper asymptotic desity 10/11 ad lower asymptotic desity 1/11 Theorem 2 For each positive iteger, the set of positive itegers with has asymptotic desity 1 + ( August September 2015] MIDDLE BINOMIAL COEFFICIENT 639

5 So it is commo for + to divide (, but what about? Theorem 3 For each iteger 0 the set of itegers > with ( is ifiite, but has upper asymptotic desity smaller tha 1 3 A remar: That there are ifiitely may itegers with ( ad also ifiitely may with ( follow from Theorems 32 ad 34 i the recet paper of Ulas [13] 6 THE PROOFS OF THEOREMS 1, 2, AND 3 There is a quic proof of Theorem 1 First assume that 2 Let p be a prime factor of ad let = where j is large eough so that > 0 I base p, has at most j digits, with the least sigificat digit beig 0 Hece, there are at most j 1 carries whe addig to ad hece + = ( For 0, let p > 2 be a odd prime umber For = p +, we have that there are o carries whe is added to itself i the base p, so that p ( But p = +, so we are doe The proof of Theorem 2 is a bit more difficult We begi with a lemma that exteds Lemma 1 to prime powers Lemma 2 Let p be a arbitrary prime, let x p be a real umber, ad (( let D = 1 + log x/ log p The umber of itegers 1 x with v p D/(5 log D is at most 3px 1 1/(5 log p Proof The calculatio here is similar to the oe i probability where you compute the chace that a coi flipped D times lads heads fairly frequetly We cosider the umber of assigmets of D base-p digits where all but at most B := D/(5 log D of them are smaller tha p/2 Sice there are p/2 itegers i [0, p/2, the umber of assigmets is at most ( D ( p/2 D j ( j p p/2 = j A crude estimatio usig D 2 gets us ( D j ( D j ( p/2 D ( p/2 D ( p p/2 D j < 2D B, ( D j p/2 so the umber of x with v p ( ( B is at most 2( p/2 D D B Now 2D B = 2e B log D 2e D/5 2e 1/5 x 1/(5 log p, p/2 2 3 p, ( 2 3 D x log(3/2/ log p, ad p D px We coclude that the umber of x with v p ( ( B is at most 2 ( p/2 D D B 2D B ( 2 3 pd 2e 1/5 px 1 log(3/2/ log p+1/(5 log p Sice 2e 1/5 < 3 ad log(3/2 >2/5, the lemma follows at oce 640 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Mothly 122 j

6 We are ow ready to prove Theorem 2 Fix a value of 1 First we claim that for p 2, (( v p v p ( + (4 Ideed, if v p ( + = j > 0, the the j least sigificat digits of + i base p are 0, so the j least sigificat digits of are at least p p/2 By Kummer s theorem, (4 holds (Note that this argumet is essetially a reprise of the proof give i Sectio 3 that Catala umbers are itegers It remais to show that for most itegers, (4 holds for all primes p < 2 Let x be a real umber ad assume that we are cosiderig values of x With D as i Lemma 2, we cosider two cases: those with v p ( + D/(5 log D ad those with v p ( + >D/(5 log D I the first case, if (4 fails, we would have v p ( ( <D/(5 log D, so by Lemma 2, the umber of beig cosidered is at most 3px 1 1/(5 log p Summig this expressio for primes p < 2 gives a quatity smaller tha 6 2 x 1 1/(5 log(2 Divided by x, this expressio teds to 0 as x,soweare left with the secod case: those with v p ( + >D/(5 log D We shall show i this case that there are very few values of x to cosider, regardless if (4 holds I fact, the umber of choices for is at most (x + /p D/(5 log D < 2x/p D/(5 log D Sice p D > p log x/ log p = x, we thus have that the umber of choices for i the secod case correspodig to the prime p < 2 is at most 2x 1 1/(5 log D Wehave log D log(1 + log x/ log 2 <1 + log log x (usig log(a + b <a/b + log b whe 0 < a < b, so that if we sum for p < 2, we get that the umber of choices for is at most 2x 1 1/(5+5 log log x Whe divided by x, this too goes to 0 as x, which completes the proof of Theorem 2 It is ot difficult to amed the proof to show that for each fixed positive iteger, the set of itegers with ( + 1( + 2 ( + ( has asymptotic desity 1 I additio, the proof allows for to ted to ifiity, provided it does ot do so too quicly i compariso with x The result (5 might be compared with Harborth [6] where it is show that for ay fixed positive iteger, almost all etries ( m j i Pascal s triagle are divisible by m(m 1 (m + 1 Also see sequeces A We ow proceed to the proof of Theorem 3, startig with the secod assertio Suppose that 0, > 2 2, ad m = has a prime factor p > The p > 2 ad writig m = cp, wehave c < m = 1 1 < 2 2 p We see that both base-p digits of = cp +, amely c ad, are smaller tha 1 p, 2 ad so there are o carries whe addig to itself i base p By Kummer s theorem, we have p ( (, so that Suppose that x > 2 4 To show the secod assertio i the theorem, it will suffice to show that whe x is sufficietly large, at least 2 x itegers 3 (22, x]havem = divisible by a prime p > 2x For each prime p satisfyig this iequality, we cout August September 2015] MIDDLE BINOMIAL COEFFICIENT 641 (5

7 umbers i (2 2, x] with p, ad this is at least (x 2 2 /p > x/p 2 No choice of correspods to two differet values of p, sice their product would be too large to have x Thus, the umber of choices for is at least 2x<p x ( x p 2 1 = x p 2π(x, 2x<p x where π(x deotes the umber of primes i [1, x] Euler proved log ago i 1737 that the sum of the reciprocals of the primes diverges to ifiity lie the double-log fuctio, ad usig a fier estimatio (such as the theorem of Mertes from 1874, we have 1 p = log log x log log( 2x + E(x, 2x<p x where E(x 0asx, see [8, 9] The differece of double logs simplifies to ( log x log 2 + log, log(2x which teds to log 2 as x As metioed before, the set of primes has asymptotic desity 0, i fact π(x/x goes to 0 lie 1/ log x by Chebyshev s estimates or the prime umber theorem Puttig these thoughts together, it follows that for each ε>0ad x sufficietly large, there are more tha (log 2 εx values of (2 2, x] with divisible by a prime p > 2x We have see that for each such umber, wehave ( Sice log 2 > > 2, the secod part of Theorem 3 follows 3 To complete the proof, we wish to show there are ifiitely may values of where ( We leave the few details for the reader, but usig Kummer s theorem, we have that if = pq + where p, q are primes with < p ad 3 p < q < 2p, the 2 ( The umber of such umbers x is greater tha a positive costat times x/(log x 2 7 THE GOVERNOR SET For each iteger, let D = { : + ( } We call D 0 the goveror set for a reaso that will soo be clear Say two sets A, B of positive itegers are asymptotically equivalet if the symmetric differece (A B \ (A B has asymptotic desity 0 I this case, we write A B For example, if A is the set of all positive itegers exceedig a googol ( ad B is the set of all composite positive itegers, the A B I particular, ay two sets of asymptotic desity 1 are asymptotically equivalet, as are ay two sets of asymptotic desity 0 If A is a set of positive itegers ad is a positive iteger, we let A + := {a + : a A} Theorem 4 For each positive iteger we have D 0 + D 642 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Mothly 122

8 We setch the proof Suppose that p is a prime with p ad p > 2 The, as i the proof of Theorem 2, v p ( ( = vp ( ( 2(+ + Ad for primes p at most 2, agai ( followig the argumet for Theorem 2, for most umbers, the power of p i both ( ad i 2(+ + is higher tha the power of p i Thus, most of the time, the coditio D 0 (that is, ( is equivalet to the coditio + D (that is, ( 2(+ + It is ot clear if the goveror set D 0 has positive lower asymptotic desity, though I cojecture this the case I Theorem 3 we essetially leared that the upper asymptotic desity is at most 1 log 2, ad perhaps some improvemet ca be made It also would be highly iterestig to ivestigate the problem umerically The umbers which divide ( are 1, 2, 6, 15, 20, 28, 42, 45, 66, 77, 88, 91, What are these umbers tryig to tell us? A cesus to higher levels may be illumiatig Note that D 0 may also be described as the set of with C(, ad i this guise there is some iformatio to be foud at sequece A To cemet the role of the goveror set D 0, let { ( } D (2 = : ( + 2 The for each positive iteger we have D (2 + D 0 The proof is similar to that of Theorem 4 So you d lie to ow how ofte the th Catala umber C( is divisible by + 1? You are agai led bac to the goveror set D 0 See too [1] ad sequece A (Notice that + 1 C( if ad oly if + 1 ( +1 This latter coditio is cosidered i [13], where i Questio 28(a, the author ass about the distributio of such Our Theorem 3 i the case = 0 is thus relevat to this query You may have oticed that ay D with 0 could have played the special role of the goveror set, but surely D 0 is the most pleasig 8 OTHER BINOMIAL COEFFICIENTS Oe might woder what the fuss is about the middle etry ( i the th row of Pascal s triagle What about oddumbered rows, where there are the twi peas ( ( +1 = +1 +1? It is possible to prove correspodig results o divisibility here For example, for 2, ( + 2( + 3 ( + usually divides ( +1 If you are iterested, you should try to prove this Looig at just the case = 2, we have a ear miss for + 2 always dividig ( +1 I fact, uless + 2 is a power of 2, it divides, ad eve i this case, it divides 2 ( +1 Sice there is a tie for the maximum etry i this row of Pascal s triagle, it maes sese to iclude both of them, ad as metioed we always have ( + 1 Thus, we might woder, as with Catala umbers, if the iteger ( (6 has combiatorial sigificace Ideed it does We ow that the Catala umber C( couts the umber of paths from (0, 0 to (, that do ot cross below the lie y = x, ad where each step of the path is oe uit to the right or oe uit up The umber i August September 2015] MIDDLE BINOMIAL COEFFICIENT 643

9 (6 is similar, but ow we are coutig paths from (0, 0 to (, + 1, a so-called ballot umber I [12], Su has the followig geeralizatio of the fact that + 1 always divides If,lare positive itegers ad every prime dividig also divides l, the ( l + 1 ( (l + for every positive iteger (7 The case = l = 1 is the situatio with Catala umbers Su ass if the coditio that every prime factor of divides l is ecessary for (7 to hold For other problems ad results cocerig divisibility properties of biomial coefficiets, the reader is referred to [3, 5, 10] ACKNOWLEDGMENT I wish to tha Sergi Elizalde, Floria Luca, ad Paul Pollac for some helpful commets I am also grateful to the referees whose careful readigs ad isightful commets improved the paper REFERENCES 1 H Balara, O the values of which mae (!/( + 1!( + 1! a iteger, J Idia Math Soc 18 ( D Bered, J E Harmse, O some arithmetic properties of middle biomial coefficiets, Acta Arith 84 ( P Erdős, R L Graham, I Z Ruzsa, E G Straus, O the prime factors of (, Collectio of articles i hoor of Derric Hery Lehmer o the occasio of his sevetieth birthday, Math Comp 29 ( R L Graham, Sequece A030979, i The o-lie ecyclopedia of iteger sequeces org 5 A Graville, Arithmetic properties of biomial coefficiets I Biomial coefficiets modulo prime powers, Orgaic Mathematics (Buraby, BC, 1995, , CMS Cof Proc, 20, Amer Math Soc, Providece, RI, H Harborth, Divisibility of ( m h by m(m 1 (m + 1, Amer Math Mothly 86 ( R D Mauldi, S M Ulam, Mathematical problems ad games, Adv Appl Math 8 ( P Pollac, Not Always Buried Deep, A Secod Course i Elemetary Number Theory Amer Math Soc, Providece, RI, , Euler ad the partial sums of the prime harmoic series, Elem Math, to appear 10 J W Sader, A story of biomial coefficiets ad primes, Amer Math Mothly 102 ( R P Staley, Eumerative Combiatorics, Vol 2 Cambridge Uiv Press, Cambridge, Z-W Su, O divisibility of biomial coefficiets, J Aust Math Soc 93 ( M Ulas, A ote o Erdős Straus ad Erdős Graham divisibility problems (with a appedix by Adrzej Schizel, It J Number Theory 9 ( CARL POMERANCE is the Joh G Kemey Parets Professor of Mathematics at Dartmouth College ad is also Research Professor Emeritus at the Uiversity of Georgia Betwee Georgia ad Dartmouth he was a member of the techical staff at Bell Labs His research is pricipally i aalytic, combiatorial, ad computatioal umber theory He is proud to be a vetera of Mississippi Freedom Summer i 1964 Dartmouth College, Haover NH carlpomerace@dartmouthedu 644 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Mothly 122