Neville Robbins Mathematics Department, San Francisco State University, San Francisco, CA (Submitted August 2002-Final Revision December 2002)

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1 Nevlle Robbs Mathematcs Departmet, Sa Fracsco State Uversty, Sa Fracsco, CA 943 (Submtted August -Fal Revso December ) INTRODUCTION The Lucas tragle s a fte tragular array of atural umbers that s a varat of Pascal s tragle I ths ote, we prove a property of the Lucas tragle that has bee merely stated by pror researchers; we also preset some apparetly ew propertes of the Lucas tragle PASCAL S TRIANGLE We beg by revewg some propertes of the tragular array of atural umbers ow as Pascal s tragle The th row of Pascal s tragle cossts of etres deoted ( ) where ad are tegers such that ad The frst 8 rows of Pascal s tragle are preseted below left-justfed format: The etres ( ) (usually called choose ) are ow as bomal coeffcets The followg propertes of bomal coeffcets are well-ow: ( ) ( )!!( )! () (symmetry) () (3) (4) ( ) (5) 4

2 odd (6) eve If, the (7) (P ascal s detty) p s prme f ad oly f (8) ( ) p p such that p I addto, there are dettes that l bomal coeffcets to Fboacc umbers, amely: F ; (9) F ; () F () Note that Pascal s tragle could be geerated ductvely usg oly (3) ad (7) The followg deftos are useful determg the hghest power of a gve prme that dvdes a bomal coeffcet Defto : If p s prme ad the teger m, let o p (m) f s the uque teger such that p m, p m Proposto : If a ad b N, the o p (ab) o p (a) o p (b) Proposto : If a, b, ad a/b N, the o p (a/b) o p (a) o p (b) Defto : If p s a prme ad the m, let the represetato of m to the base p be gve by: m r a p where a p, a r Defto 3: Wth p ad m as Defto, let t p (m) deote the sum of the dgts of m to the base p, that s r t p (m) a 43

3 Proposto 3: If p s prme ad, the (( )) o p t p() t p ( ) t p () p Proposto 4: If m, the ( ) m s eve for all such that m Proposto 5: If m, the ( ) m s odd for all such that m Remars: Propostos ad follow easly from Defto Proposto 3 follows from 4 Theorem 35, p 54 (See also 4, exercses 3-3, p 55-56) Propostos 4 ad 5 follow from Proposto 3 3 THE LUCAS TRIANGLE The Lucas tragle s a fte tragular array of atural umbers whose th row cossts of etres that we wll deote, where ad The symbol may be defed ductvely as follows: ; () If, the (3) The frst eght rows of the Lucas tragle are preseted below, left-justfed format: Note that whereas Pascal s tragle s geerated by the coeffcets of (a b), the Lucas tragle s geerated by the coeffcets the expaso of (a b) (a b) Below, we lst some propertes of the Lucas tragle Note that (3), (9), (), (), () are aalogues of (7), (4), (5), (6), (8) respectvely (4) 44 f ad (5)

4 eve p (6) (7) ( ) f (8) 3( ) (9) ( ) f () 3( ) f () odd p s prme f ad oly f () p such that p Remars: Idetty (4) follows from (), (3), (3), (7), ad ducto o Each of (5), (6), (7), (8) follow from (4); (9) follows from (5) ad (4), whle () follows from (5) ad (5), as we shall demostrate below Idetty () follows from (9) ad (), whle () s Theorem We ote that (5), (8), ad (9) were stated wthout proof I addto, the followg dettes l to Fboacc ad Lucas umbers: L ; (3) F ; (4) F (5) Note that (3), (4), (5) are aalogues of (9), (), () respectvely 45

5 4 NEW RESULTS We beg by provg detty (3), whch has bee prevously hted dagrammatcally ad stated wthout proof 3 Just as the sums of rsg dagoals Pascal s tragle yeld the Fboacc umbers, so do the rsg sums of dagoals the Lucas tragle yeld the Lucas umbers Theorem : If, the L Proof: (Iducto o ) We ote that L so the statemet holds for, Now ad L 3, L L L by ducto hypothess Therefore, we have L If m, the we have Now (3) mples If m, the we have L m L m L m m m m ( m m m m m m m m m m ) m m m m m m 46

6 Now (3) mples L m m ( m m m ) m m m The proofs of dettes (4) ad (5) are smlar, ad are therefore omtted Next, we prove detty () Theorem : ( ) f Proof: Ivog (8) ad (5), we have ( ) ( ) ( ) j ( ( ) ( ) ( ) ( ) j j ( )) The ext theorem cocers rsg dagoals the Lucas tragle ad s somewhat remscet of detty (): Theorem 3: If p s a odd prme, the ( ) p o p { f p f Proof: Idettes (7) ad () mply p p ( ) p p < p p(p )! (!)(p )! Sce p by hypothess, t follows that o p (p(p )!) Now { f p o p ((!)(p )!) o p ((p )!) p f < p 47

7 The cocluso ow follows from Proposto The followg theorem descrbes a row property ejoyed by odd prmes Theorem 4: If p s a odd prme, the for all j such that j p ad for all such p j that j p, we have p Proof: (Iducto o j) Idettes (7) ad () mply p p ( ) p (p )(p )! (!)(p )! Sce p, t follows that p dvdes the umerator, but ot the deomator of the latter fracto Therefore the theorem holds for j Now (3) mples that p j p j p j By the ducto hypothess, each of the summads of the rght member s dvsble by p Therefore the left member s dvsble by p, so we are doe The fal theorem cocers the party of Lucas tragle etres rows such that the row umber s a power of m Theorem 5: s odd for all such that m 5 Proof: It suffces to voe (8) wth m, ad the mae use of Propostos 4 ad REFERENCES Mar Feberg A Lucas Tragle Fboacc Quart 5 (967): H W Gould & W E Greg A Lucas Tragle Prmalty Test Dual to That of Ma- Shas Fboacc Quart 3 (985): Verer E Hoggatt Jr A Applcato of the Lucas Tragle Fboacc Quart 8 (97): , 47 4 N Robbs Begg Number Theory (993) Wm C Brow Publshers (Dubuque, IA) AMS Classfcato Numbers: B65, B39 48