#A63 INTEGERS 11 (2011) SUBPRIME FACTORIZATION AND THE NUMBERS OF BINOMIAL COEFFICIENTS EXACTLY DIVIDED BY POWERS OF A PRIME

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1 #A63 INTEGERS 11 (2011) SUBPRIME FACTORIZATION AND THE NUMBERS OF BINOMIAL COEFFICIENTS EXACTLY DIVIDED BY POWERS OF A PRIME William B. Everett Cherogolovka, Moscow Oblast, Russia bill@chget.ru Received: 9/4/10, Revised: 7/26/10, Accepted: 9/29/11, Published: 10/10/11 Abstract We use the otio of subprime factorizatio to establish recurrece relatios for the umber of biomial coefficiets i a give row of Pascal s triagle that are divisible by p j ad ot divisible by p j+1, where p is a prime. Usig these relatios to compute this umber ca provide sigificat savigs i the umber of computatioal steps. 1. Itroductio Subprime factorizatio coverts the prime divisibility of Pascal s triagle of biomial coefficiets ito a set of subprime-divisibility triagles. Each subprime-divisibility triagle is a very simple regular tilig with two basic triagular tiles: oe tile cotaiig oly zeros ad the other cotaiig oly oes. Two differet subprimedivisibility triagles for a prime p differ oly i scale ad resolutio, ad the relative scale ad resolutio factor is a power of p. This set of regular tiligs of Pascal s triagle provides the most detailed iformatio about the prime divisibility of biomial coefficiets i a simple form that allows easily obtaiig various geeral results. Let be a oegative iteger ad p be a prime. Let θ j (, p) deote the umber of biomial coefficiets ( ) k, 0 k, such that p j ( ) k, i.e., p j ( ad p j+1 (. We represet i the base p: = c0 +c 1 p+c 2 p 2 + +c r p r, 0 c i < p, i = 0, 1,..., r, c r > 0 for 0. For j = 0 ad p = 2, Glaisher, i 1899 [4], established the formula θ 0 (, 2) = 2 r r i=0 ci = (c i + 1). Fie geeralized this result to ay prime p i 1947 [3]: θ 0 (, p) = r i=0 (c i + 1). For j = 1, Carlitz gave the specific formula r 1 θ 1 (, p) = (c 0 + 1) (c i 1 + 1)(p c i 1)c i+1 (c i+2 + 1) (c r + 1) i=0 i 1967 [1]. Howard gave formulas for θ j (, 2) ad for θ 2 (, p) i 1971 ad 1973 [5, 6]. We recetly gave the followig geeral formula for θ j (, p) [2]. i=0

2 INTEGERS: 11 (2011) 2 We let W be the set of r-bit biary words, i.e., W = { w = w 1 w 2... w r : w i {0, 1}, 1 i r }, ad partitio W ito r+1 subsets W j, 0 j r, where each subset cosists of all the words cotaiig exactly j oes: { r } W j = w W : w i = j. We sum over the words i the subset W j to obtai θ j (, p): i=1 θ j (, p) = r 1 F (w)l(w) M(w, i), (1) w W j i=1 where the fuctios F (w), L(w), ad M(w, i) givig the r+1 factors i each summad are defied as { c if w 1 = 0, F (w) = p c 0 1 if w 1 = 1, c r + 1 if r > 0 ad w r = 0, L(w) = c r if r > 0 ad w r = 1, 1 if r = 0, c i + 1 if w i = 0 ad w i+1 = 0, p c i 1 if w i = 0 ad w i+1 = 1, M(w, i) = c i if w i = 1 ad w i+1 = 0, p c i if w i = 1 ad w i+1 = 1. We take b i=a (... ) = 1 if a > b. Oversights i formula (1) i [2] i the cases r = 0, 1 are corrected here. The sum i (1) cotais ( r j) terms, which meas that to compute the complete set of θ j (, p), 0 j r, the umber of computatioal steps as a fuctio of r (which, of course, depeds o ad p) basically icreases as r ( r j=0 j) = 2 r. We obtai recurrece relatios that allow computig the set of θ j (, p) with the umber of steps basically icreasig as r 2. To simplify obtaiig values of θ j (, p) i practice, it is coveiet to ot calculate the idividual values separately but treat the sequece {θ 0 (, p), θ 1 (, p),... } as a etity. We let capital letters deote sequeces, for example, A = {a 0, a 1,... } ad B = {b 0, b 1,... }. We have the usual operatios of a liear space over itegers. Obviously, k(a ± B) = ka ± kb. We also defie a shifted sequece: Obviously, A = { a 0 = 0, a 1 = a 0, a 2 = a 1,... }. (A±B) = A ± B ad (ka) = k A.

3 INTEGERS: 11 (2011) 3 Let Θ(, p) = {θ 0 (, p), θ 1 (, p),... }. For the recursio base, we have two lowest value cases: < p (i.e., = c 0 ) or p < 2p (i.e., = c o + p). For r = 0, ad for r = 1 ad c 1 = 1, Θ(, p) = {c 0 + 1, 0, 0,... }, (2) Θ(, p) = {2(c 0 + 1), p c 0 1, 0, 0,... }. (3) I Sectio 4, we prove the followig theorems. Theorem 1 (Geeral recurrece relatios). For r > 1 ad c r = 1, Θ(, p) = 2Θ( p r, p) + (p c r 1 ) Θ( p r (c r 1 1)p r 1, p) ad for r > 0 ad c r > 1, (p c r 1 + 1) Θ( p r c r 1 p r 1, p), (4) Θ(, p) = c r Θ ( (c r 1)p r, p ) (c r 1)Θ( c r p r, p). (5) Theorem 2 (Special recurrece relatio). If = p k m 1 (i.e., the k least sigificat digits i the base-p represetatio of are p 1), the Θ(, p) = p k Θ(m 1, p). (6) Remark. If θ j (, p) is desired for oly oe value of j, the we ca write the recurrece relatios i compoet form with the covetio that θ j (, p) 0 for j < 0. For (4) ad (5), we have θ j (, p) = 2θ j ( p r, p) + (p c r 1 )θ j 1 ( p r (c r 1 1)p r 1, p) (p c r 1 + 1)θ j 1 ( p r c r 1 p r 1, p), r > 1, c r = 1, θ j (, p) = c r θ j ( (cr 1)p r, p ) (c r 1)θ j ( c r p r, p), r > 0, c r > 1. The recurrece relatio θ 0 (, p) = 2θ 0 ( p r, p) follows obviously from Fie s formula for j = 0 with c r = 1, but the recurrece relatios i Theorem 1 are less obvious for j > 0. Remark. Our motivatio for seekig the geeral recurrece relatios preseted here was Shevelev s otio of biomial predictors, which provides a recursive defiitio of Θ(, p) for a very special class of [9]. Special recurrece relatio (6) geeralizes Shevelev s result ad is applicable to 1/p of all. The biomial predictor otio, i additio to strogly restrictig the least sigificat digit(s) i the base-p represetatio of exactly as i the applicability coditio for (6), also restricts the more sigificat digit(s). As a example, we cosider Example 3 i [9]: p = 3, = 23, θ 0 (23, 3) = 18, θ 1 (23, 3) = 6. Here is the secod member of

4 INTEGERS: 11 (2011) 4 the sequece 7, 23, 71, 215,... for which we have the correspodig sequeces of θ 0 6, 18, 54, 162,... ad of θ 1 2, 6, 18, 54,.... This, for istace, immediately suggests cosiderig the sequece 6, 20, 62, 188,..., for which we have the correspodig sequeces of θ 0 3, 9, 27, 81,... ad of θ 1 4, 12, 36, 108,.... The situatio with the sequece of biomial predictors (begiig with 7) is prettier tha the situatio with the secod sequece. With the biomial predictors, we ca read the θ values directly from the base-p represetatio of + 1. I both cases, if we kow the θ values for the first member of the sequece, the we obtai the θ values for ay subsequet member by multiplyig by the appropriate power of p. This suggested the existece of a more geeral recurrece relatio. Remark. While we had supposed that Kummer s theorem o carries [7] could be used to obtai geeral formula (1) preseted i [2] ad the recurrece relatios i Theorem 1 i this paper, the otio of subprime factorizatio leads directly ad obviously to the geeral formula ad the recurrece relatios. After the first draft of this paper was completed, we became aware of [8], where a geeral formula was directly obtaied usig Kummer s theorem. The formula for a p α, the umber of ozero etries modulo p α i row of Pascal s triagle, give i [8] is essetially α 1 j=0 θ j(, p) i our otatio. I Sectio 2, we explai the otio of subprime factorizatio ad idicate how it results i geeral formula (1). This otio simplifies the problem of fidig a geeral formula for the umber of biomial coefficiets exactly divided by a fixed power of a prime ad might be applicable or adaptable to similar problems. Moreover, it is cetral i provig the recurrece relatios. I Sectio 3, we give a few simple umerical examples usig the recurrece relatios. I Sectio 4, we prove the recurrece relatios. I the coclusio, we briefly discuss the relative computatioal efficiecy of the methods for determiig the umber of biomial coefficiets exactly divided by a fixed power of a prime. 2. Subprime Factorizatio We used the otio of subprime (subscripted prime symbol) factorizatio to obtai geeral formula (1). The first few atural umbers ca be represeted with ordiary prime factorizatio as 1, 2, 3, 2 2, 5, 2 3, 7, 2 3, 3 2. With subprime factorizatio, we ca represet these umbers as 1, 2 1, 3 1, , 5 1, , 7 1, , We emphasize that a subprime is ot a umber ad we do ot actually perform arithmetic operatios with subprimes. This otio seems uecessary ad useless whe lookig at the sequece of atural umbers, but it is quite useful whe cosiderig the questio of how may biomial coefficiets are exactly divided by a give power of a prime. The questio reduces to cosiderig which subprimes appear i the subprime factorizatio of the biomial coefficiets because there are o powers of subprimes. Ad elimiatig powers drastically simplifies the problem.

5 INTEGERS: 11 (2011) 5 The followig facts are well kow ad easily established. These properties are more ofte stated i terms of Pascal s triagle modulo a prime. The statemet p ( ( ad = 0 mod p are obviously equivalet. P1. If < p, the p ( for ay k. If p is a prime factor of!, the p. For < p, p ( would the imply the cotradictio p < p. P2. For 0 < k < + 1, if p ( ( k 1) ad p ) ( k, the p +1 ) ( k, ad if p k 1) ad p ( ) ( k, the p +1 ) k except if + 1 is a multiple of p. The first statemet holds because multiplicatio is distributive over additio. The secod statemet is equivalet to if ( ( k 1) 0 mod p ad 0 mod p ad ( +1 ) ( = k 0 mod p, the + 1 = 0 mod p. Now, ( k 1) 0 mod p ad 0 mod p ad ( ) ( +1 = k 0 mod p implies ( = k 1) a mod p ad = p a mod p for some a, 0 < a < p. But such a pair of adjacet biomial coefficiets occurs i Pascal s triagle modulo p oly i a row for which = p 1 mod p. P3. If is a multiple of p i for i > 0, i.e., = mp i, 0 < m p, the p ( for all 0 < k < such that k is ot a multiple of p i except if is a power of p, i which case p ( for all 0 < k <. This property ca be established by direct calculatio of the powers of p i the prime factorizatio of!, k!, ad (! i the relevat cases for m < p ad m = p. We take correspodig statemets to defie the subprime factorizatio of the biomial coefficiets. The basic differeces are that p is replaced with p 1... p i i the coditio of the first fact ad that divides ad is a multiple are replaced with appears i the subprime factorizatio of ad the subprime factorizatio of cotais. S1. If < p 1... p i, the p i does ot appear i the subprime factorizatio of ( k for ay k. S2. For 0 < k < + 1 ad + 1 ot a multiple of p, p i appears i the subprime factorizatio of ( ) +1 k if ad oly if pi appears i the subprime factorizatios of both ( ( k 1) ad. S3. If the subprime factorizatio of cotais p 1... p i, the p i appears i the subprime factorizatio of ( for all 0 < k < such that p1... p i does ot appear i the subprime factorizatio of k with o exceptios. The absece of exceptios i S2 ad S3 simplifies thigs. We cosider represetatios of Pascal s triagle. I the limit of a ifiite umber of rows, Pascal s triagle of biomial coefficiets modulo a prime is self-similar; for )

6 INTEGERS: 11 (2011) 6 p = 2, it has the fractal dimesio log 2 3 (see, e.g., [10]). The prime divisibility fuctio (( )) { 1 if ( f p = = 0 mod p, k 0 if ( 0 mod p, maps the self-similar Pascal s triagle modulo p ito the correspodig self-similar p-divisibility triagle. Passig from prime to subprime factorizatio coverts the self-similarity to self-cogruecy i a sese. The fractal object becomes a set of regular tiligs. Takig p = 2 as a example, the first few rows of the 2-divisibility triagle are Row Here, the rows 0 ad 1 correspod to P1, eve umbered rows correspod to P3, ad the other rows correspod to P2. The correspodig subprime-divisibility triagles for 2 1, 2 2, ad 2 3 are Row

7 INTEGERS: 11 (2011) 7 I these subprime-divisibility triagles, the rows umbered 0 through 2 i 1 follow from S1, rows that are multiples of 2 i follow from S3, ad all other rows follow from S2. We give a simple example of observatios that ca follow from the subprime factorizatio of Pascal s triagle. If we replace p i p 1 p 2... p i for all primes i the factorizatio of the atural umbers, the 2 1 divides every secod umber exactly oce, 2 2 divides every fourth umber exactly oce, 2 3 divides every eighth umber exactly oce (here ad hereafter, we use terms like divide as shorthad for appears i the subprime factorizatio of ). This does ot seem to gai us much beyod elimiatig the eed to keep track of the powers i the ordiary prime factorizatio. But if we defie the triagular umbers as the sequece of biomial coefficiets ( 2), = 2, 3,..., the we see from the 2i -divisibility triagles that 2 1 does ot divide ay triagular umber, that 2 2 divides the triagular umbers for of the forms 4m ad 4m + 1, that 2 3 divides the triagular umbers for of the forms 8m ad 8m + 1, ad so o. It is already visually obvious that p i for p + i > 3 divides triagular umbers thus defied for of the forms p i m ad p i m + 1. Ad it is clear why 2 1 is the oly exceptio. The same kid of reasoig ca be applied to the sequece of tetrahedral umbers defied as ( 3), = 3, 4,..., ad so o. We ca view the subprime-divisibility triagles as tiligs with a triagle A of zeros ad a triagle V of oes as the two basic tile types. Viewed as triagles cosistig of zeros ad oes, the subprime-divisibility triagles differ for differet subprimes (ad for differet primes), but they are basically idetical whe viewed as tiligs. The kth tile row, k 0, cosists of a triagle A followed by k copies of the rhombus formed by cocateatig a triagle V ad a triagle A. If we take oe elemet (0 or 1) as the uit legth, the the tiligs differ i scale (the size of the tiles). If we take a characteristic tile legth (p i for example) as the uit legth, the the tiligs oly differ i resolutio (elemets per uit legth). If we plotted the triagle for 2 9 at a scale of oe ich equals the tile legth with a white pixel for 0 ad a black pixel for 1, the the lie betwee the triagles A ad V would look very much like a straight lie at 512 ppi (pixels per ich). Moreover, if we plotted the similar triagle for 5 4 at a scale of oe ich equals the tile legth, the it would be difficult to distiguish the picture at 512 ppi from the picture at 625 ppi. I other words, we ca view the subprime-divisibility triagles as all the same tiligs except for their resolutio. I the limit as p i, the basic tiles form a perfect square with the lower-left white triagle equal i area to the upperright black triagle (we take the diagoal of the square to belog to the lower-left triagle, but this lie has o area). I the discrete case (the digitized picture), the triagle A has the area p i k=1 k ad the triagle V has the area p i 1 k=1 k (ad the differece i area is the umber p i of elemets o the diagoal). Lettig deote cocateatio, we ca write R = V A for the rhombus formed by cocateatig a triagle V ad a triagle A. Similarly, kr, for example, deotes

8 INTEGERS: 11 (2011) 8 the cocateatio of k copies of R. We idicate the size of of these objects with subscripts. For example, A p i is the triagle A i the subprime-divisibility triagle for p i, also called the p i -divisibility triagle. The row legths (umber of zeros) of A p i from top to bottom rage from 1 to p i. The row legths (umber of oes) of the triagle V p i from top to bottom rage from p i 1 to 0 (it is coveiet to allow a row of zero legth). Viewig the p i -divisibility triagle as a tilig, we ca easily see that the regio from p i to p i+1 cotais p tile rows A p i kr p i, 0 k < p. The kth tile row i this regio overlays the rows of Pascal s triagle umbered by kp i < (k + 1)p i. Furthermore, it is easily see that A p i+1 ad V p i+1 respectively correspod to p tile rows of the p i -divisibility triagle as A p i kr p i ad (p k 1)R p i V p i, 0 k < p. As a example, we show the tiles A 5 ad V 5 (with omitted subscripts) i the rage 25 < 50 with the separatios betwee the correspodig A 25 ad V 25 idicated by spaces ad with the rage of show for each tile row: 25 to 29 A VAVAVAVAV A 30 to 34 AVA VAVAVAV AVA 35 to 39 AVAVA VAVAV AVAVA 40 to 44 AVAVAVA VAV AVAVAVA 45 to 49 AVAVAVAVA V AVAVAVAVA. This example is obviously geeralizable to ay prime p: we simply have p tile rows ad p V triagles i the first of those rows. We ow cosider some simple summaries of the iformatio cotaied i the set of subprime-divisibility triagles for a give prime p. Treatig the elemets zero ad oe of the subprime-divisibility triagles as Boolea truth values ad sum these triagles usig Boolea additio, the we obtai the self-similar prime-divisibility triagle. If we treat the elemets of the subprime-divisibility triagles as ordiary itegers ad sum these triagles usig ordiary additio, we obtai a triagle of the degrees of the highest powers of the prime dividig the biomial coefficiets. For example, summig the subprime-divisibility triagles for 2 i, i = 1, 2, 3, we obtai Row

9 INTEGERS: 11 (2011) 9 Formula (1) is a slightly more complicated summary of iformatio from the set of subprime-divisibility triagles. We illustrate the statemet at the begiig of this sectio that the questio of how may biomial coefficiets are exactly divided by a give power of a prime reduces to cosiderig which subprimes appear i the subprime factorizatio of the biomial coefficiets. We give some specific terms F (w)l(w) r 1 i=1 M(w, i) i the summatio i (1) for large r ad j = 2: for w = , the term (p c 0 1)(c r + 1)(p c 1 )c 2 (c 3 + 1) (c r 1 + 1) couts the umber of ( that cotai p1 ad p 2 ad o other p i (this term is zero if c 0 = p 1 or c 2 = 0); for w = , the term (p c 0 1)(c r + 1)c 1 (p c 2 1)c 3 (c 4 + 1) (c r 1 + 1) couts the umber of ( that cotai p1 ad p 3 ad o other p i (this term is zero if c 0 = p 1 or c 1 = 0 or c 2 = p 1 or c 3 = 0); ad for w = , the term (c 0 + 1)(c r + 1)(p c 1 1)(p c 2 )c 3 (c 4 + 1) (c r 1 + 1) couts the umber of ( that cotai p2 ad p 3 ad o other p i (this term is zero if c 1 = p 1 or c 3 = 0). Addig these terms does ot strictly make sese from the subprime stadpoit (it is like addig a cout of apples ad a cout of orages ad a cout of pears). But passig from subprime factorizatio back to ordiary prime factorizatio allows summig the terms (somewhat like cosiderig apples, orages, ad pears to be fruits). I summary, the otio of subprime factorizatio simplifies problems cocerig powers of primes that divide biomial coefficiets by, first, passig from primes ad powers of primes to a ifiite set of subprimes without powers (idetical except for scalig) with each subprime associated with a specific power of the prime ad, secod, passig from a self-similar fractal object to a ifiite set of regular tiligs (idetical except for scalig ad resolutio). Such a approach might be applicable or adaptable to other problems. 3. Examples We give a few simple examples to help i visualizig the formulas. We take p = 5 ad cosider a few four-digit umbers (r = 3) i base 5. We ote that the Θ sequeces have a fiite positive portio: there exists a oegative iteger k such that θ k (, p) > 0 ad θ j (, p) = 0 for all j > k. For brevity i the umerical examples, we let θ 0 (, p),..., θ k (, p) deote the fiite positive portio of the sequece Θ(, p).

10 INTEGERS: 11 (2011) 10 Example. = 586 = To determie Θ(586, 5) Θ( ), we must calculate Θ( ), Θ(321 5 ), Θ(121 5 ), ad Θ(21 5 ) usig the geeral recurrece relatios ad Θ(11 5 ) ad Θ(1) usig the base formulas. From the base formulas, we obtai Θ(11 5 ) (3) = 4, 3, Θ(1) (2) = 2. From the geeral recurrece relatios, we obtai the additioal itermediate values Θ(21 5 ) (5) = 2Θ(11 5 ) Θ(1) = 2 4, 3 2 = 6, 6, Θ(121 5 ) (4) = 2Θ(21 5 ) + 3 Θ(11 5 ) 4 Θ(1) = 2 6, , 4, 3 4 0, 2 = 12, 16, 9, Θ(321 5 ) (5) = 3Θ(121 5 ) 2Θ(21 5 ) = 3 12, 16, 9 2 6, 6 = 24, 36, 27, Θ( ) (4) = 2Θ(321 5 ) + 2 Θ(121 5 ) 3 Θ(21 5 ) We the obtai = 2 24, 36, , 12, 16, 9 3 0, 6, 6 = 48, 78, 68, 18. Θ( ) (5) = 4Θ( ) 3Θ(321 5 ) = 4 48, 78, 68, , 36, 27 = 120, 204, 191, 72, i.e., θ 0 (586, 5) = 120, θ 1 (586, 5) = 204, θ 2 (586, 5) = 191, ad θ 3 (586, 5) = 72. Example. = 156 = I additio to the same two base values as i the above example, we must calculate the itermediate value We the obtai Θ(111 5 ) (4) = 2Θ(11 5 ) + 4 Θ(11 5 ) 5 Θ(1) = 2 4, , 4, 3 5 0, 2 = 8, 12, 12. Θ( ) (4) = 2Θ(111 5 ) + 4 Θ(111 5 ) 5 Θ(11 5 ) = 2 8, 12, , 8, 12, , 4, 3 = 16, 36, 57, 48, i.e., θ 0 (156, 5) = 16, θ 1 (156, 5) = 36, θ 2 (156, 5) = 57, ad θ 3 (156, 5) = 48. Example. = 149 = We ote that 149 = Therefore, we ca use the special recurrece relatio Θ(149, 5) (6) = 25Θ(5, 5). I this case, we obtai Θ( ) (6) = 25Θ(10 5 ) = 25 2, 4 = 50, 100, i.e., θ 0 (149, 5) = 50 ad θ 1 (149, 5) = 100.

11 INTEGERS: 11 (2011) Proofs of the Recurrece Relatios I provig Eqs. (2) (6), i additio to the A p i otatio i Sectio 2, we use subscripts idicatig the the first ad last rows overlaid by the object. For example, A 2pi,3p i 1 = A p i. We also use the otio of a restricted Θ(, p) object cotaiig the couts for the portio of the th row of Pascal s triagle overlaid by the object. Thus, for p < 2p, we ca write Θ(, p) Θ(, p) Ap V p A p = Θ(, p) Ap + Θ(, p) Vp + Θ(, p) Ap = 2Θ(, p) Ap + Θ(, p) Vp. Proof of base formula (2). Base formula (2) follows directly from fact P1 i Sectio 2 because the umber of biomial coefficiets i the th row of Pascal s triagle is + 1. Proof of base formula (3). For umbers = p + c 0, we have Θ(, p) = 2Θ(, p) Ap,2p 1 + Θ(, p) Vp,2p 1. Because A p,2p 1 = A 0,p 1, the first term i the right-had side is twice the sequece give by base formula (2). From the discussio i Sectio 2, p divides every biomial coefficiet overlaid by V p,2p 1 V p. Therefore, the first term i Θ(, p) Vp must be zero. No p i divides ay ( for i > 1 because < p i. Therefore, oly the secod term i Θ(, p) Vp is ozero, ad the value of this term must be +1 2(c 0 +1) = p c 0 1. Base formula (3) is proved. Proof of geeral recurrece relatio (4). We first examie the objects that overlay the rows of Pascal s triagle from p r to 2p r 1. These rows are overlaid by A pr,2p r 1 V pr,2p r 1 A pr,2p r 1. From the discussio i Sectio 2, Θ(, p) Ap r,2p r 1 is clearly idetical to Θ( p r, p). Hece, Θ(, p) = 2Θ( p r, p) + Θ(, p) Vp r,2p r 1. We examie V pr,2p r 1 i relatio to the subprime-divisibility triagles. The portio of the triagle of the degrees of the highest powers of p dividig the biomial coefficiets overlaid by V pr,2p r 1 cotais the arithmetic sums of the correspodig portios of the subprime-divisibility triagles for p 1 through p r 1 plus oe (V pr,2p r 1 cotais oly oes i the subprime-divisibility triagle for p r ). The plus oe for all the powers of p i this overlaid regio correspods to the shift operatio o sequeces. This regio cotais p r 1 rows correspodig to (p 1)R p r 1,2p r 1 1 V p r 1,2p r 1 1 followed by p r 1 rows correspodig to (p 2)R p r 1,2p r 1 1 V p r 1,2p r 1 1,

12 INTEGERS: 11 (2011) 12 ad so o dow to the last p r 1 rows correspodig to V p r 1,2p r 1 1. The top set of rows correspods to p r < p r + p r 1, the secod set correspods to p r + p r 1 < p r + 2p r 1, ad so o. Altogether, the p sets of p r 1 rows correspod to p r < 2p r. We ow have Θ(, p) Vp r,2p r 1 = (p c r 1 1) Θ( p r (c r 1 1)p r 1, p) Rp r 1,2p r Θ( p r (c r 1 1)p r 1, p) Vp r 1,2p r 1 1. (7) We ow seek the restricted Θ i terms of urestricted Θ. Obviously, Θ( p r (c r 1 1)p r 1, p) = Θ( p r (c r 1 1)p r 1, p) Ap r 1,2p r Θ( p r (c r 1 1)p r 1, p) Rp r 1,2p r 1 1 = 2Θ( p r (c r 1 1)p r 1, p) Ap r 1,2p r Θ( p r (c r 1 1)p r 1, p) Vp. r 1,2p r 1 1 Moreover, we have Θ( p r (c r 1 1)p r 1, p) Ap = Θ( p r c r 1,2p r 1 r 1 p r 1, p). 1 Therefore, ad Θ( p r (c r 1 1)p r 1, p) Rp r 1,2p r 1 1 = Θ( p r (c r 1 1)p r 1, p) Θ( p r c r 1 p r 1, p) Θ( p r (c r 1 1)p r 1, p) Vp r 1,2p r 1 1 = Θ( p r (c r 1 1)p r 1, p) 2Θ( p r c r 1 p r 1, p). Substitutig the expressios with urestricted Θ i (7) ad combiig like terms, we obtai geeral recurrece relatio (4). Proof of geeral recurrece relatio (5). It is ow easy to prove geeral recurrece relatio (5). Clearly, for r > 0 ad c r > 1, Θ(, p) = Θ( (c r 1)p r, p) Ap r,2p c r Θ( (c r 1)p r, p) Rp r,2p 1 1 = (c r + 1)Θ( (c r 1)p r, p) Ap r,2p c r Θ( (c r 1)p r, p) Vp r,2p 1 1. (8) Similarly to the precedig proof, we have Θ( (c r 1)p r, p) Vp r,2p 1 1 = Θ( (c r 1)p r, p) 2Θ( c r p r, p)

13 INTEGERS: 11 (2011) 13 ad Θ( (c r 1)p r, p) Ap r,2p 1 1 = Θ( c r p r, p). Substitutig the expressios with urestricted Θ i (8) ad combiig like terms, we obtai geeral recurrece relatio (5). Proof of special recurrece relatio (6). It is clear from the discussio i Sectio 2 that the th row, p i 1, of the p i -divisibility triagle cotais oly zeros if ad oly if p i + 1. It follows that for p i+1 1, if the th row of the p i -divisibility triagle is ot all zeros, the the th row of the p i+1 -divisibility triagle is ot all zeros. Because θ r (, p) is a cout of the umber of positios cotaiig 1 i the th rows of all the subprime-divisibility triagles for p 1 through p r, θ r (, p) > 0 if ad oly if p + 1. Similarly, θ r 1 (, p) > 0 if ad oly if p Ad so o. It remais to show that if θ r (, p) = 0, the p θ j (p, ) for 0 j < r, ad this easily follows by iductio o r. Notig that the applicability coditio = p k m 1 for special recurrece relatio (6) is equivalet to p k + 1, we have proved relatio (6). Remark. From the foregoig, it is clear that if θ j (, p) = 0, the θ k (, p) = 0 for all k j. Moreover, θ 1 (, p) = 0 if ad oly if has the form cp s 1, where 0 < c < p ad s is a oegative iteger; θ 2 (, p) = 0 if ad oly if has the form c r p r + cp r 1 1, where r > 1; ad so o. 5. Coclusio We briefly cosider the computatioal efficiecy of differet ways of computig Θ(, p) by lookig at the basic rate of icrease i the umber of computatioal steps as a fuctio of icreasig r. Traditioally, Θ(, p) has bee computed by coutig the umber of carries whe addig k ad k i base p (Kummer s formula for the degree of the highest power of p that divides ( [7]). To compute Θ(, p), we sum 1 pairs (there is o poit i summig for ( ( 0) ad ) ). This meas summig c r p r + + c 0 1 pairs, ad the basic rate of icrease is as p r. Geeral formula (1) for θ j (, p), which follows because the subprime factorizatio of Pascal s triagle of biomial coefficiets gives essetially the same regular biary tilig for all powers of all primes ad the tiles oly differ i scale ad resolutio, reduces the basic rate of icrease from p r to r i=0 ( r i) = 2 r. Usig the geeral recurrece relatios i Theorem 1 i the geeral case with = c 0 +c 1 p+ +c r p r, we calculate oe value for Θ(c 0, p), two values for Θ(c 0 +p, p), two values for Θ(c 0 +c 1 p, p),..., r+1 values for Θ(c 0 +c 1 + +p r, p), ad r+1 values for Θ(c 0 + c 1 p + + c r p r, p). I all, the umber of itermediate values calculated before calculatig the fial r +1 values is r 2 +2r 1. The umber of computatioal steps for calculatig each value is obviously bouded by a small positive costat

14 INTEGERS: 11 (2011) 14 idepedet of r. Therefore, the basic rate of icrease i computatioal steps is as r 2, which is a improvemet over 2 r. Refereces [1] L. Carlitz. The umber of biomial coefficiets divisible by a fixed power of a prime. Red. Circ. Mat. Palermo (2), 16 (1967), [2] W. B. Everett. Number of biomial coefficiets divisible by a fixed power of a prime. Itegers, 8, (2008), A11. [3] N. J. Fie. Biomial coefficiets modulo a prime. Amer. Math. Mothly, 54 (1947), [4] J. W. L. Glaisher. O the residue of a biomial-theorem coefficiet with respect to a prime modulus. Quart. J. Math., 30 (1899), [5] F. T. Howard. The umber of biomial coefficiets divisible by a fixed power of 2. Proc. Amer. Math. Soc., 29 (1971), [6] F. T. Howard. Formulas for the umber of biomial coefficiets divisible by a fixed power of a prime. Proc. Amer. Math. Soc., 37 (1973), [7] E. E. Kummer. Über die Ergäzugssätze zu de allgemeie Reciprocitätsgesetze. J. Reie Agew. Math., 44 (1852), [8] E. S. Rowlad. The umber of ozero biomial coefficiets modulo p α. arxiv: v2. [9] V. Shevelev. Biomial coefficiet predictors. J. Iteger Sequeces, 14 (2011), ; Biomial predictors. arxiv: v4. [10] S. Wolfram. Geometry of biomial coefficiets. Amer. Math. Mothly, 91 (1984), mathematics/84- geometry/1/text.html.