MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES. Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang, Selangor, Malaysia
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1 Malaysan Journal of Mahemacal Scences 9(2): (2015) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homeage: h://ensemumedumy/journal A Mehod for Deermnng -Adc Orders of Facorals 1* Rafka Zulkal, 1 Kamel Arffn Mohd Aan and 1,2 S Hasana Saar 1 Insue for Mahemacal Research, Unvers Pura Malaysa, UPM Serdang, Selangor, Malaysa 2 Dearmen of Mahemacs, Faculy of Scence, Unvers Pura Malaysa, UPM Serdang, Selangor, Malaysa E-mal: rafkazulkal@gmalcom, kamel@umedumy and shas@umedumy *Corresondng auhor ABSTRACT In hs aer, wh a rme he adc sze of n! where n s a osve neger s deermned for n 0 and n 0 The dscusson egns wh he deermnaon of he adc szes of facoral funcons!, q! and q! wh, 0 and q a rme dfferen from I s found ha! 1 1 wh 0 Resuls are hen used o oan he exlc form of adc szes of n from works of earler auhors I s also found ha he adc ers of n C r n! n r! r! s gven y ln ln C where n and r wh 0 Keyws: Facoral funcons, adc szes
2 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar 1 INTRODUCTION In hs aer, we resen a mehod for deermnng adc ers of n!, for any ove neger n We use he noaon x, where s a rme and x s any raonal numer o denoe he hghes ower of dvdng x We refer o x as he adc er or he adc sze of x I follows ha, for wo raonal numers of x and y, x xy x y, x y and y x y mn x, y By convenon x f x 0 The noaon x wll as usual denoe he greaes neger funcon Wh n! where n general r!n r! n r, n C r wll denoe he quoen x! x x 1 x Lengyel, 2003 dscussed he er of lacunary sums of nomal k n k coeffcen of he form Gn, lk 0 l n where n ndcaes he facoral r n! funcon Hs sudy nvolves negers n of he form n r!n r! where s an odd rme and 0 Adelerg, 1996 examned he adc n ers of n! and where n r and esalshed some new congruence r relaons assocaed wh adc neger er Bernoull numers He aled he relaons o rove he rreducly roery of ceran Bernoull olynomals wh ers ha are dvsle y Wagsaff, 1996 dscussed he Aurfeullan facorzaons and he erod of he Bell numers modulo a rme He showed ha exonenal negers 1 s he mnmum erod modulo of he Bell Malaysan Journal of Mahemacal Scences
3 A Mehod for Deermnng -Adc Orders of Facorals Kolz, 1977 deermned he adc szes of n! where n can e exressed as a numer n ase rme numer n he form 2 s n Sn n a0 a1 a2 as ha s n! wh Sn a 1 he summaon of he coeffcen n n for 0 a 1 Berend, 1997 saed ha here exs nfnely many neger osve n such ha n!!! 0 mod 2 1 n 1 k n where 1, 2,, k are rme facorsn n! n ascendng er Laer, Yong, 2003 mroved he resul oaned y Sander, 2001 and showed ha here exs nal values of n for any rme facorzaon of n! Suose k s any neger and 0,1 for 1, 2,, k I s shown ha here exs nfnely many osve neger n wh n! mod 2, n! mod 2,, n! mod k , Yong and We roved ha f q s rme and l, any osve negers hen lq! l q! l! q q q Based on he works of Kolz, 1997 and Mohd Aan and Loxon, 1986, Saar and Mohd Aan, 2002 examned he coeffcens of lnear aral dervave olynomals f and f assocaed wh he quadrac olynomal 2 2, x y f x y ax xy cy dx ey m and gave he esmae of he adc szes of her common zeros n erms of he adc ers of he coeffcens Laer n 2007, hey examned he coeffcens of a olynomal o arrve a he adc esmae of common zeros of k In h 6 degree f and n erms of he adc ers of he coeffcens n he domnan erms of f x, y For a olynomal n he nomal form, he coeffcens are exressle n erms of he facorals Tha s, n n n n n n! f x, y ax y C ax y where C Such cases n!! 0 necessae a mehod o deermne he adc ers of he facorals n n C We egn our dscusson wh Secon 20 for deermnng he adc ers of! and q! where, q rmes wh q and 0 We also derve a formula for adc ers of q! wh, 0 n hs secon In he susequen Secon 30, we resen a mehod for deermnng x f y Malaysan Journal of Mahemacal Scences 279
4 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar k he adc szes of n! where n k wh k 0 for n 0 and n 0 A he end of hs secon, we dscuss he fndngs of revous researchers regardng he deermnaon of adc szes for any facorals In he followng Secon 40, we resen a mehod for deermnng he adc szes of he facoral funcon n C where n and r wh 0 Ths s wh a vew o deermne he adc szes of coeffcens of he erms n he exanson of he olynomal f x, y ax y where n s osve n he cases n 2 -ADIC ORDERS OF FACTORIAL FUNCTIONS Le n e a osve neger, and q rmes We wll frs consder and n q where qand, 0 In hs secon, we wll deermne he adc szes of! and q! followed y he adc szes of q! wh, Adc Orders of α! and β q! We egn our dscusson y nroducng a lemma for deermnng he numer of facors k n! where 0 k 1 such ha k where a non-negave neger as follows Lemma 211 Suose s a rme, 0, 1 k 1 and Then, here exs such ha k 1 facors r k n! 1 1 Proof Le! k Then! k Le k 0 1 k0 e a facor n! k0 such ha k where 280 Malaysan Journal of Mahemacal Scences
5 A Mehod for Deermnng -Adc Orders of Facorals The numer of such facors s he same as he numer of k such ha k snce clearly k Now, k when k wh 0 for some neger Hence,, k, I follows ha Now, 1 k mles ha The numer of such ha 1 k, 1 1 Tha s 1 and Euler oen funcon, 1 1 1, 1, s gven y he 1 1 k Our asseron follows 1 Hence, here exs values of k, 1 k 1 such ha Corollary 211 If s a rme, 0, 1 k 1 and 0 1, hen 1 1! Proof Le! k k Then! k k k1 k1 Le e a non-negave neger By Lemma 211, for each here exs 1 1 facors k where 1 k 1 n! such ha k Therefore,! Malaysan Journal of Mahemacal Scences 281 1
6 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar Tha s 0 1 1! 1 Ths corollary wll e aled for he roof of he followng heorem: Theorem 211 Le e a rme and 0 Then 1! 1 Proof By Corollary 211, 1! k1 k2 1 2 k k On smlfyng we oan! Malaysan Journal of Mahemacal Scences
7 A Mehod for Deermnng -Adc Orders of Facorals Nex, we nvesgae he adc szes of! where q s a rme and q and 0 A frs, we nroduce he followng lemma ha wll e used for he nex heorem For a osve neger n we deermne he numer of negers m where 0 m n, whose adc ers are non-vanshng Lemma 212 Le e a rme, n 0 and m s an neger wh n 0 mn Then here exs negers m such ha m 0 Proof Snce m s an neger and m 0, m s of he form m k where k 0 Now, le S m m k, k 1,2,3, e he se of negers m such ha k 0 Then, k gves he numer of negers m such ha m 0 Now, n 0 m n mles ha 0 k n Tha s, 0 k n Snce k s an neger, 0 k n Therefore, S m m k, k 1,2,3,, n n Clearly, here exs elemens n S I follows ha, here exs negers m such ha m 0 We can now aly hs lemma for deermnng he numer of facors q k n q! whose adc ers are non-negave, as follows: Theorem 212 Le q q Then, here exs 1 q ln q k where 0 ln q, q e any rme and q, 0 and 0 k q 1 facors q k n q! such ha Malaysan Journal of Mahemacal Scences 283
8 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar q 1 Proof Le q! q k Then q! q k k 0 For each facor q k n! q k when q 1 k 0 q k where 0 Now, q, le q k m where m 0 Thus, he numer of such facors of q! s gven y he numer of negers m such ha m 0 for every We wll deermne hs numer as follows: Clearly, k q m Snce 0 k, we have 0 q m q from q q q whch 0 m Snce m s an neger, 0 m By Lemma 212, q here exs 1 negers m such ha m 0 Therefore, here exs q q 1 negers m such ha m 0 Hence, he numer of facors q k q q n! 1 q such ha q k s gven y q Snce m s an neger wh1 m, we have q Therefore, ln q 0 ln We wll recover he resul of Lemma 211 y leng q n he frs ar of he aove heorem The followng heorem gves he adc er of q! where qand 0 Theorem 213 Le q, e any rme, q and 0 Then q ln q ln q q 1 0! 284 Malaysan Journal of Mahemacal Scences
9 A Mehod for Deermnng -Adc Orders of Facorals q 1 Proof Le q! q k Then q q k k 0 q 1! Le e a non-negave neger By Theorem 212, for each here exs q q 1 facors q k n q! such ha k, where ln q 0 Thus, q ln 22 Adc Orders of k0 ln q ln q q 1 0! q! Theorem 221 gves he adc ers of q! wh, 0 usng he resul from Theorem 213 Theorem 221 Suose and q rme wh ln q ln 1 q q! q q Then q 1 Proof From he defnon, q! q k Ths equaon can e rewren as k0 q,, 0 and 0 1 q 1 q 1 q! q j q wh 0 j0 1, ndcang 1 1 q 1 q q 1 Now, q j q j q! j0 j0 Malaysan Journal of Mahemacal Scences 285
10 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar q 1 wh 1 q 1 Therefore, q! q! q Then 1, q 1 1 q 1!! q q q Snce q 1 1, q , q q! q, q 0 for 1 q 1 I follows ha 1 1 q 0 Thus, q q q 1,!! From hs equaon, y relacng we have 1 1 q! q q! where Therefore, q! q q q q q q! q! q Summng he geomerc rogresson, we oan 1 q! q q! 1 1 q! q q! 1 Tha s, 286 Malaysan Journal of Mahemacal Scences
11 A Mehod for Deermnng -Adc Orders of Facorals From Theorem 213, wh q, we have q ln q ln q q 1 0! I follows ha ln q ln 1 q q! q q 3 -ADIC ORDERS OF n! Le e any rme and n a osve neger In hs secon, we resen our man resuls on deermnng he adc szes of n! where n s exressed n s rme ower decomoson of he form k n k wh 0 for 1,2,, k In er o deermne he adc szes of n!, we need o consder he value of n whch can e dvded no wo cases They are negers n such ha n 0 and n 0 31 Adc Orders of n! wh n 0 In hs secon, we dscuss he adc ers of n! wh n 0 as n he followng heorem n1 From he defnon, n! n k Therefore, n! n k k0 n1 k0 Suose n k s a facor n n! wh 0 k n The followng lemma and heorem show ha he adc szes of n! deends on he numer of hese facors The resul of Lemma 212 s used n he roof of Lemma 311 as follows: Lemma 311 Suose s a rme and n e a osve neger wh n 0 and 0 k n Le e a non-negave neger Then, here exs Malaysan Journal of Mahemacal Scences 287
12 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar n n 1 ln n 0 ln facors n k n! n such ha n k wh Proof n1 From he defnon, n! n k, n! n k k0 n1 Suose n k s a facor n n! wh k0,1,2,, n 1 and a nonnegave neger Now, n k when Thus, he numer of such facors of n k0 n k m wh m 0 k such ha s gven y he numer of negers m m 0 for each Now, consder he facors n n k m n k n n! k such ha n k Thus where m 0 Snce k 0 and k n m, we have 0 m n n n Snce m s an neger, 0 m, here exs negers m for every k such ha m 0 n From Lemma 212, here exs 1 n n follows ha here exs 1 every negers m such ha m 0 I negers m such ha m 0 for Thus, he numer of such facors n k wh k0,1,2,, n 1 n n! such n k s gven y n n 1 for every ha 288 Malaysan Journal of Mahemacal Scences
13 A Mehod for Deermnng -Adc Orders of Facorals n Snce m s an neger wh 1m, we have n Hence, ln n 0 ln The followng Theorem 311 gves he adc szes of n! for n 0 usng he resul from Lemma 311 Theorem 311 Suose s a rme, n a osve neger wh n 0 and a non-negave neger Then, n! n1 ln n ln n n 1 0 Proof Snce n! n k, we have n! n k k0 n1 k0 n n From Lemma 311, here exs 1 k0,1,2,, n 1 n! follows ha n! n such ha ln n ln 1 0 n k wh facors n k wh ln n 0 ln I n n 32 Adc Orders of n! wh n 0 In hs secon, we resen he case n whch he adc ers of n! s osve The followng heorem gves a resul on he adc szes of such n! y usng resul from Theorem 311 Theorem 321 Le e any rme and n a osve neger such ha n where 0 Then, Malaysan Journal of Mahemacal Scences 289
14 Proof Gven n Clearly, n1 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar ln n ln 1 n n n! n n wh 0 Then, n n1 wh n From he defnon, n! n1! n1 k n k0 The roduc on he rgh hand sde of he equaon can e rewren as a roduc of wo facors accng o he adc szes of k, whch are k 0 and k 0 n 1 n Thus, n! n1! n1 j n1 where ha 0 n 1 n 1 j0 1, n1 1 Now, n1 j n1 j n1! j0 j0 n1 1 1 n1 1 Therefore, n1! n1! n1 1, n 1 Then n1! n1! n1 1 1 n1 ndcaes 1 1, 1 1 Tha s, n1! n1 n1! n1 Snce n 11 1,, n1 0 for 1 n1 1 n 11 n1 0 1, I follows ha 290 Malaysan Journal of Mahemacal Scences
15 A Mehod for Deermnng -Adc Orders of Facorals Thus, n! n n! (1) Le e an neger n he range 1 1 Then y relacng y Therefore y (1) and (2), n Equaon (1) we would have 1 1 n! n n! (2) n! n! n n n + + n + + n n 1 n1! n 1 n! 1 1 Hence, n! n n 1! 1 On smlfyng, we oan n! n1 n1! 1 Snce n1 0, from Theorem 311, we have n ln n1 ln n1 n1 1! 1 0 Malaysan Journal of Mahemacal Scences 291
16 Thus, Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar ln n1 ln 1 n1 n1! n n Leng n 1 ln n ln 1 n n follows ha n! n n, The followng corollary of Theorem 321 shows he adc ers of n! where n wh 0 Corollary 321 Suose n s any osve neger wh rme ower k decomoson n k Le n n 1,2,, k Then, ln n ln 1 n n! n n for k Proof Gven n k wh 0 for 1,2,, k Clearly, n 0 As n he roof of Theorem 321, we oan 1 n! n! n 1 wh n 0 for 1,2,, k Snce n 0, y Theorem 311 we have ln n ln n n! 1 0 n Hence, ln n ln 1 n n! n n wh n n 292 Malaysan Journal of Mahemacal Scences
17 A Mehod for Deermnng -Adc Orders of Facorals Kolz, 1977 showed ha f s any rme and n osve neger where n s exressed as a numer n ase rme of he form 2 s n Sn n a0 a1 a2 as and 0 a 1, hen n! 1, he summaon of he coeffcens of n n, 0 where Sn a comarson, Theorem 311 and Theorem 321 gve n! ln n ln n n for 0 n and 1 0 ln n ln 1 n n n! n for n 0 resecvely Boh formulae gve alernave ways of deermnng n! ased on ceran condons wh he laer whou havng o exress he value of n n ase rme numer By In he revous fndngs of adc szes of arcular facorals y Yong and We, 2007, s shown ha for all rmes q and any osve negers and l, eq lq! l eqq! eq l! The noaon eq xreresens he q adc er of an neger x Based on Theorem 221, y nerchangng rmes, q and ln ln q q 1 1, we oan q q! 1 q 1 0 q q where q and 0 Ths gves he q adc szes of q! for any rme and q Now, we aly our mehod o oan exlc resul for q lq! for any osve neger l We need only o deermne he value of q l! snce ha of q! s readly avalale from Theorem 211 In q er o evaluae q l!, here are wo cases o consder; hey are he case when q l 0 and q l 0 Malaysan Journal of Mahemacal Scences 293
18 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar The followng heorem gves he l 0 usng he resul from Theorem 311 q q adc szes of lq! wh q l and Theorem 322 Suose q s any rme and l, are osve negers wh q l and l 0 Suose e a non-negave neger Then q lnl ln q q 1 l l q! 1 q 1 0 q q lq l Proof Yong and We, 2007 showed ha lq! l q! l! q q q wh 0 From Theorem 211, wh q a rme and 0, we have q 1 q q! q 1 As well as from Theorem 311 wh q l 0, we have Thus, ln l ln q l l q l! 1 0 q q lnl ln q q 1 l l q lq! l 1 q 1 0 q q Now, he nex heorem gves he q l, 0 usng he resul from Theorem 321 q adc szes of lq! wh q l and Theorem 323 wh q l Suose q s any rme and and l are osve negers 294 Malaysan Journal of Mahemacal Scences
19 Suose A Mehod for Deermnng -Adc Orders of Facorals q l wh 0 and s a non-negave neger Then, ln lq ln q q 1 l l q! 1 q q 1 0 q q lq l Proof From he works of Yong and We, 2007, s shown ha q lq! l q q! q l! wh 0 From Theorem 211, wh q rme and 0, we oan q 1 q q! q 1 As well as from Theorem 321, wh lnlq ln q q l q 1 l l q l! l 1 q q 1 0 q q Then, lnlq ln q, 0, we have q 1 q 1 l l q lq! l l 1 q 1 q q 1 0 q q Tha s, ln lq ln q q 1 l l q lq! l 1 q q 1 0 q q 4 -ADIC ORDERS OF n C r Le n, r e negers wh n r In hs secon, we wll dscuss he n n! adc szes of Cr, for he case n and r Clearly r! n r! snce n r Now, C!!! Therefore, C!!! Malaysan Journal of Mahemacal Scences 295
20 By Theorem 211, we have Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar 1 1 C 1 1!! 1 (3) The value of! s deermned y he followng Theorem 411 Frs, we have he followng asseron: Lemma 411 Suose s a rme, 0 and 0 k Then, here exs facors k n ln! such ha k where 0 ln 1 Proof Le! k 1 k0 Then, k! k0 Le k e a facor n! where 0 Now, such ha k k when k m where m 0 Thus, he numer of such facors n! s gven y he numer of negers m such ha m 0 for every Now, k m Tha s, k 0,1,2,, 1 m k wh 296 Malaysan Journal of Mahemacal Scences
21 Thus, 0 m A Mehod for Deermnng -Adc Orders of Facorals Snce m s neger hen 0 m By Lemma 212, here exs m negers m such ha Therefore, here exs m negers m such ha Hence, he numer of facors of k n! such ha k s gven y 1 1 Snce m s an neger and 1 m, we have Therefore, ln 0 ln Theorem 411 Le e any rme, 0, hen Proof ln ln 1 1! 0 1 1! k k k0 k0 Malaysan Journal of Mahemacal Scences 297
22 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar From Lemma 411, here exs k n! ln 0 ln Thus, 1 1 such ha k ln ln facors, where 1 1! 0 We nex deermne he value of C as follows: Theorem 412 Le e a rme, 0, hen ln ln C Proof The roof follows from equaon (3) and Theorem CONCLUSION In hs aer, we have resened a mehod for deermnng he adc szes of n! where n s a osve neger and s a rme The resuls oaned are n exlc forms and he mehod of oanng hem offers an alernave way o fndng n! As resened n hs aer, he mehod does no requre he neger n o e exressed n ase as s usually done I also enales one o oan more exlc resuls of adc szes of lq! where l s an neger, q a rme and 0 To llusrae alcaon of resuls oaned n hs aer, adc szes of n C where n r wh 0 are deermned Ths mehod s exendale o r and 298 Malaysan Journal of Mahemacal Scences
23 A Mehod for Deermnng -Adc Orders of Facorals deermnng n C r for any osve n and r The adc szes of oher exressons conanng facoral facors may also e found y alyng he resuls n hs aer ACKNOWLEDGEMENT We would lke o acknowledge wh hanks he fnancal suor for our research from Fundamenal Research Gran Scheme of Malaysa REFERENCES Adullah, A R (1991) The four on Exlc Decouled Grou (EDG) mehod: A fas Posson solver Inernaonal Journal of Comuer Mahemacs 38: Adelerg, A (1996) Congruences of -Adc Order Bernoull Numers Journal of Numer Theory 59: Berend, D (1997) On he Pary of Exonens n he Facorzaon of n! Journal of Numer Theory 64: Kolz, N (1977) -Adc Numers, -Adc Analyss and Zea Funcons NewYork: Srnger-Verlag Lengyel, T (2003) On he Order of Lacunary Sums of Bnomal Coeffcens Elecronc Journal of Comnaoral Numer Theory 31:10-12 Mohd Aan, K A and J H Loxon (1986) Newon Polyhedra and Soluons of Congruences In Loxon, JH and Van der Pooren, A(ed) Dohanne Analyss Camrdge : Camrdge Unversy Press Sander, J W (2001) On he Pary of Exonens n he Prme Facorzaon of Facorals Journal of Numer Theory 90: Saar, S H, and K A Mohd Aan (2002) Penganggaran Kekardnalan Se Penyelesaan Persamaan Kongruen Jurnal Teknolog 36(C) : Malaysan Journal of Mahemacal Scences 299
24 Rafka Zulkal, Kamel Arffn Mohd Aan & S Hasana Saar Saar, S H, and K A Mohd Aan (2007a) Penganggaran Saz -Adc Pensfar Seunya Teran Seara Polnomal Berdarjah Enam Sans Malaysana 36(C) : Saar, S H, and K A Mohd Aan (2007) A Mehod of Esmang -Adc Szes of Common Zeros of Paral Dervave Polynomals Assocaed Wh an n h Degree Form Malaysan Journal of Mahemacs Scences 1(1) : Wagsaff, Jr S S (1996) Aurfeullan Facorzaons and he Perod of he Bell Numers Modulo a Prme Mahemacs of Comuaon 65: Yong, G C (2003) On he Pary of Exonens n he Sandard Facorzaon of n! Journal of Numer Theory 100: Yong, G C and We, L (2007) On he Prme Power Facorzaon of! n (Par 2) Journal of Numer Theory 122: Malaysan Journal of Mahemacal Scences
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