An analog of the arithmetic triangle obtained by replacing the products by the least common multiples

Transcription

1 arxiv: v2 [mathnt] 9 Feb 2010 A aalog of the arithmetic triagle obtaied by replacig the products by the least commo multiples Bair FARHI bairfarhi@gmailcom MSC: 11A05 Keywords: Al-Karaji s triagle; Least commo multiple; Biomial coefficiets 1 Itroductio The Al-Karaji arithmetic triagle is the triagle cosistig of the biomial coefficiets ( (, N, Precisely, for each N, the th row of that triagle is: ( ( (, 0 1 where ( :! ( 1 ( +1!(! 1 2 So the begiig of the arithmetic (or biomial triagle is give by: (1 Note that the costructio of the triagle rests o the property that each umber of a give row is the sum of the umbers which are situated just above Explicitly, 1

2 we have: ( ( ( 1 (, such that 1 (2 Historically, the first mathematicia who discovered the biomial triagle was the pioeer arabic mathematicia Al-Karaji ( AD He drew this triagle util its 12 th row ad oted the process of its recursive costructio by poitig out (2 More iterestigly, Al-Karaji discovered the biomial formula: (x+y 0 ( x y ( N (3 After Al-Karaji, several other mathematicias of the Islamic civilizatio reproduced that very importat triagle (Al-Khayyam, Al-Samawal, Al-Tusi, Al-Farisi, Ib Al-Baa, Ib Muaim, Al-Kashi, The same triagle have bee discovered agai i Chia (Yag Hui i the 13 th cetury I Europ (16 th cetury, several mathematicias remared the importace of Al-Karaji s triagle (Stifel, Tartaglia, Pascal, I this paper, we are goig to obtai the aalog of Al-Karaji s triagle by substitutig i Formula (1 the products by the least commo multiples If we use the formula ( would be:!!(!, the lcm-aalog of the biomial coefficiet ( lcm(1,2,, lcm(1,2,, lcm(1,2,, But this aalogy is ot quite iterestig because those last umbers are ot all itegers For example, for 6, 3, we have: lcm(1,2,,6 lcm(1,2,3 lcm(1,2,3 5 3 Z I order to obtai a iterestig aalogy, we will use rather the formula ( ( 1 ( +1 So, the lcm-aalog of a biomial coefficiet ( 1 2 which we must cosider is: [ ] lcm(, 1,, +1 : (4 lcm(1,2,, (We aturally covetioe that lcm( 1 Notice that a table of the umbers [ ] was already give by A Murthy (2004 ad exteded by E Deutsch (2006 i the O-Lie Ecyclopedia of Iteger Sequeces (see the sequece A of OEIS However, to my owledge, o property was already proved about those umbers i compariso with their aalog biomial umbers 2

3 2 Results We begi with the easy result showig that the ratioal umbers [ ], defied by (4, are all itegers We have the followig: Propositio 1 For all atural umbers, such that, the positive ratioal umber [ ] is a iteger Proof Let, be atural umbers such that Amog the cosecutive itegers, 1,, + 1, oe at least is a multiple of 1, oe at least is a multiple of 2,, ad oe at least is a multiple of This implies that lcm(, 1,, +1 is a multiple of each of the positive itegers1,2,, Cosequetly lcm(, 1,, +1 is a multiple of lcm(1,2,,, which cofirms that [ ] is a iteger The propositio is proved Defiitio Throughout this paper, we call the umbers [ ]: the lcm-biomial umbers ad we call the triagle cosistig of them: the lcm-biomial triagle The begiig of the lcm-biomial triagle is give i the followig: (Here the colored umbers i gree are those that are differet from their aalog biomial umbers Now, we are goig to establish less obvious results cocerig the lcmbiomial umbers Theorem 2 For all atural umbers, such that, the lcm-biomial umber [ ] divides the biomial umber ( Proof Actually the theorem ca be immediately showed by usig a result of S Hog ad Y Yag [3] which states that for all itegers, (with 0, 1, the positive iteger g (1 divides the positive iteger g (, where g deotes 3

4 the Farhi arithmetical fuctio 1 (see Lemma 24 of [3] But i order to put the reader at their ease, we give i what follows a idepedet ad complete proof Let, N such that 1 ad The statemet of the theorem is clearly equivalet to the followig iequalities: (( ([ ] v p v p (for all prime umber p (5 (where v p deotes the usual p-adic valuatio Let us show (5 for a give prime umber p O the oe had, we have: (( (! v p v p!(! v p (! v p (! v p ((! α1 ( α1 α1 α1 (where represets the iteger part fuctio It is importat to stress that each of the terms ( (α 1, of the last sum, is oegative ideed, for all positive iteger α, we have: + p + α But sice + is a iteger, the we have eve: +, which cofirms the stressed fact Now, o the other had, we have: ([ ] ( lcm(, 1,, +1 v p v p lcm(1,2,, a b, where a : v p (lcm(, 1,, +1 ad b : v p (lcm(1,2,, (6 1 By defiitio: g ( : (+1 (+ lcm(,+1,,+ (, 4

5 Note that because [ ] is a iteger (accordig to Propositio 1, we have a b By defiitio, a is the greatest expoet α of p for which divides at least a iteger of the rage (,] Sice for all α N, the umber of itegers belogig to the rage (,], which are multiples of, is exactly equal to, the we have: a max { α N : } 1 Similarly, b is (by defiitio the greatest expoet α of p for which divides at least a iteger of the rage [1,] But sice for all α N, the umber of itegers belogig to the rage [1,], which are multiples of, is exactly equal to, the we have: { b max α N : } 1 ( Remarig that the sequece is o-icreasig (sice each α N of the terms represets the umber of itegers lyig i the rage (,], which are multiples of, we have: α N,α a : 1 Further, from the defiitio of b, we have: Cosequetly, we have: α N (b,a] : α N,α > b : 0 1 Accordig to (6, it follows that: (( ( v p α1 ( b<α a 1 b<α a a b v p ([ ], (7 (8 5

6 which cofirms (5 ad completes this proof Now, by Theorem 2, we see that the ratios ( /[ ] are actually positive itegers But it certaily remais several other profoud properties to discover about those umbers We ca as for example about the couples (, satisfyig the equality ( [ ] The followig theorem shows a very importat property for the ratios ( /[ ] We derive from it for example that for a fixed colum, the umbers ( ( /[ ] lie i a fiite set of positive itegers ( ( Theorem 3 For all N, the sequece of positive itegers [ ] ad its smallest period T is give by: T p prime, p < where 0 if v p ( max v p(i 1 i< α p max v p(i otherwise 1 i< p, As a importat cosequece, we derive the followig: is periodic ( p prime,p < Corollary 4 For( all N, the positive iteger lcm(1, 2,, 1 is a period ( of the sequece [ ] Admittig Theorem 3, the proof of Corollary 4 becomes obvious: it suffices (( to remar that the exact period T, give by Theorem 3, of the sequece /[ ] clearly divides lcm(1,2,, 1 To prove Theorem 3, we use the arithmetical fuctiosg ( N itroduced by the author i [1] ad studied later by Hog ad Yag [3] ad by Farhi ad Kae [2] For a give N, the fuctio g is defied by: g : N\{0} N\{0} g ( : (+1 (+ lcm(,+1,,+ I [1], it is just remared that g is periodic ad that! is a period of g The Hog ad Yag [3] improved that period to lcm(1, 2,, ad recetly, Farhi ad Kae [2] have obtaied the exact period of g which is give by: P p prime, p p 0 if v p ( +1 max 1 i v p (i max 1 i v p (i otherwise 6

7 Kowig this result, the proof of Theorem 3 becomes easy: Proof of Theorem 3 For a fixed N, a simple calculus shows that for ay N, we have: ( [ g 1( +1 ] g 1 (1 This last idetity clearly shows that for ay give N, the sequece (( /[ ] is periodic ad that its exact period is equal to the exact period of g 1 So by the Farhi-Kae theorem, the exact period of (( /[ ] is P 1, as claimed i Theorem 3 We ed this sectio by givig the lcm-biomial triagle util its 12 th row The lcm-aalog of Al-Karaji s triagle Note that The lcm-biomial umbers colored i gree are those that are differet from their aalog biomial umbers 3 Some remars ad ope problems about the lcm-biomial umbers 1 Ca we prove Theorem 2 without use prime umber argumets? 2 Describe the set of all the couples (, ( 0 satisfyig [ ] ( 7

8 3 Let N Sice for ay {0,1,,}, we have [ ] ( (because [ ] divides (, accordig to Theorem 2 the for all oegative real umber x, we have: [ ] ( x x (1+x, that is: 0 0 [ 0 ] x (1+x ( x 0 (9 Taig x 1 i (9, we deduce i particular that for all N, we have [ /2 ] 2 (where deotes the ceilig fuctio But sice [ /2 ] lcm(, 1,, /2 +1 is a iteger (accordig to Propositio 1, lcm(1,2,, /2 the lcm(, 1,, /2 +1 is a multiple of lcm(1,2,, /2 Cosequetly we have lcm(, 1,, /2 +1 lcm(, 1,, /2 + 1;1,2,, /2 lcm(1,2,, So [ /2 ] 2 gives: lcm(1,2,, 2 lcm(1,2, /2 ( N The iteratio of the last iequality gives: lcm(1,2,, 2 + /2 + / log 2 ( 4 ( 1 Hece: lcm(1,2,, 4 ( 1, which is a otrivial upper boud of lcm(1,2,, The questio which we pose is the followig: Ca we more judiciously use Relatio (9 to prove a otrivial upper boud for the least commo multiple of cosecutive itegers that is sigificatively better tha the previous oe? 4 It is easy to see that ufortuately there is o a iteral compositio law of N which satisfies for ay positive itegers, ( : [ ] [ ] [ ] (the aalog of (2 Ideed, if we suppose that such a law exists the we would have o the oe had [ 2 1 ] [2 2 ] [3 2 ], that is ad o the other had [4 3 ] [4 4 ] [5 4 ], that is 2 1 5; which gives a cotradictio The problem which we pose is the followig: 8

9 Fid a iterative costructio (ie, a costructio row by row for the lcm-biomial triagle 5 For a give positive iteger d, let Ω(d deote the umber of prime factors of d, coutig with their multiplicities I this item, we loo at the diagoals of the lcm-biomial triagle We costat that the first diagoal (which we ote by D 0 cotais oly the 1 s; i other words, we have: d D 0 : Ω(d 0 0 The secod diagoal (oted D 1 is cosisted oly o the 1 s ad the prime umbers; i other words, we have: d D 1 : Ω(d 1 Also, the third diagoal of thelcm-biomial triagle (otedd 2 is cosisted of positive itegers havig at most two prime factors (coutig with their multiplicities; i other words, we have: d D 2 : Ω(d 2 More geerally, we have the followig: Propositio 5 For N, let D deote the ( + 1 th diagoal of the lcm-biomial triagle The, we have: d D : Ω(d The proof of this propositio is actually very easy ad leas oly o the followig simple fact: { N : lcm(1,2,,,+1 p if +1 is a power of a prime p lcm(1,2,, 1 otherwise Proof of Propositio 5 Let N fixed ad let d D So, we ca write d as: d [ + ] lcm(+1,+2,,+ (for some N It follows that lcm(1,2,, d divides the positive iteger lcm(1,2,,+ But we costat that the last lcm(1,2,, lcm(1,2,,+i umber is the product of the positive itegers (1 i lcm(1,2,,+i 1 each of which is either a prime umber or equal to 1 (accordig to the fact metioed just before this proof So, it follows that: ( lcm(1,2,,+ Ω(d Ω lcm(1,2,, The propositio is proved 9

10 Note that by usig prime umber theory, we ca improve the obvious upper boud of Propositio 5 to: d D : Ω(d c log, where c is a absolute positive costat (effectively calculable Refereces [1] B Farhi Notrivial lower bouds for the least commo multiple of some fiite sequeces of itegers, J Number Theory, 125 (2007, p [2] B Farhi & D Kae New results o the least commo multiple of cosecutive itegers, Proc Am Math Soc, 137 (2009, p [3] S Hog & Y Yag O the periodicity of a arithmetical fuctio, C R Acad Sci Paris, Sér I 346 (2008, p