EXTENSION OF THE GCD STAR OF DAVID THEOREM TO MORE THAN TWO GCDS CALVIN LONG AND EDWARD KORNTVED

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1 EXTENSION OF THE GCD STAR OF DAVID THEOREM TO MORE THAN TWO GCDS CALVIN LONG AND EDWARD KORNTVED Astrt. The GCD Str of Dvi Theorem n the numerous ppers relte to it hve lrgel een evote to shoing the equlit of the gretest ommon ivisors of to sets of elements forme prtitioning vrious rrs of inomil oeffiients for n lotion of these rrs in Psl s tringle. In this pper e eten the stu to rrs ivie into n susets ith n equl gretest ommon ivisors for n = 2 n ultimtel for ritrr n Introution The genesis of Goul s Str of Dvi Conjeture [] first prove Hillmn n Hoggtt [] s pper Hoggtt n Hnsell [] shoing tht the prout of the si inomil oeffiients surrouning n given entr in Psl s tringle is perfet squre. Equivlentl if one numers onseutivel the si entries forming hegon roun n given entr the prout of the even numere entries equls the prout of the o numere entries. In ition to Goul s onjeture Hoggtt n Hnsell s pper sprke numer of ppers [ ] on perfet squre n equl prout ptterns in Psl s tringle. The strting point from the present pper is the pper Usiskin [11] in hih he shoe tht the prout of the s equls the prout of the s equls the prout of the s for the three-imon rr of inomil oeffiients shon in Figure 1 for n lotion of the rr in Psl s tringle. Figure 1. No s Goul s le to onjeture the originl GCD Str of Dvi Theorem onsiering the pper Hoggtt n Hnsell e sho here tht g (S ) = g (S ) = g (S ) here S S n S re the sets of s s n s in Figure 1 n g (S t ) enotes the gretest ommon ivisor of the elements in S t for t = or. In ft similr results n e otine for n n--n imons for n = 2 n n e onjeture tht similr results hol for ll greter vlues of n. In the lst setion e sho ho to onstrut ritrril lrge tringulr rrs tht n e prtitione into n susets ith equl GCDs for ritrr n 2. The proofs of ll these results epen on the folloing esil prove lemms [1 ]. 12 VOLUME NUMBER

2 GCD STAR OF DAVID THEOREM TO MORE THAN TWO GCDS Lemm 1. For the three rrs shon if n re jent oeffiients nhere in Psl s tringle then g ( ) = g ( ). Lemm 2. For the to rrs shon if n re oeffiients nhere in Psl s tringle ith n jent to then g ( ) = g ( ). 2. Results for Dimons We first onsier Usiskin s rr of Figure 1. Theorem 1. For n lotion of the rr of oeffiients in Figure 1 in Psl s tringle g (S ) = g (S ) = g (S ). Proof. We first numer the entries in Figure 1 s shon in Figure r t s u v Figure 2. Let = g (S ) e = g (S ) n f = g (S ). Sine i for 1 i repete use of Lemm 1 n Lemm 2 sho tht t 1 r s u v 2 1 n 2. Therefore e n f. Similrl one shos tht e n e f n lso tht f n f e. Thus = e = f s lime. Results entirel nlogous to Theorem 1 ut for n = 2 n n ll e prove using onl Lemm 1 n Lemm 2. The hllenge in eh se of ourse is to prtition the rr uner onsiertion into susets for hih the gretest ommon ivisors re inee equl. Also e note tht e ere unle to get similr results for n imons ith n though e hve little out tht suh results re true for ritrr n. Intereste reers n ontt the uthors for suitle prtitions for n = 2 n. NOVEMBER

3 THE FIBONACCI QUARTERLY. Results for Tringles of Lest Support It is interesting to note tht Lemm 1 n Lemm 2 immeitel llo for the etension of Theorem 1 to ht might e lle the equilterl tringle of lest support of n rr; i.e. the smllest equilterl tringle ontining the rr in question here the top sie is long ro of Psl s tringle n hose other to sies re long min igonls s shon for the to imons in the figure hih is the to imon nlogue of Figure 1. Also note tht eh ot in the figure n e reple either or ithout hnging the finl result. Further e oserve tht Lemm 1 n Lemm 2 llo the etension of ll theorems in the eisting literture shoing equl GCDs to e etene to soli tringles of lest support for the rrs in question here gin the e oeffiients n e llote to n of the susets into hih the rr in question s originll ivie ithout ltering the result.. A Generl Constrution for Aritrril Mn GCDs For ritrr n 2 e give onstrution for ritrril lrge tringulr sets of inomil oeffiients tht n e prtitione into susets S i 1 i n ith the propert tht g (S i ) = g (S j ) for 1 i < j n for n positioning of the onstrute set in Psl s tringle. The onstrution is ompletel generl n oul e given generl proof. Hoever to onserve spe e onsier onl the se n =. Consier the rr of oeffiients shon in Figure Figure. Theorem 2. For the rr of Figure n n tringulr rr otine from it etening the pttern ritrril fr to the right lote nhere in Psl s tringle g (S ) = g (S ) = g (S ) = g (S ). 1 VOLUME NUMBER

4 GCD STAR OF DAVID THEOREM TO MORE THAN TWO GCDS Proof. Let = g (S ) e = g (S ) f = g (S ) n g = g (S ). First onsier. As rgue erlier Lemm 1 n Lemm 2 suffie to sho tht ivies ll the elements in the equilterl tringle of oeffiients of lest support ontining n. Moreover it is ler tht n element of S must pper t lest one in the rr mong the top five elements of eh igonl in the rr sloping onr n to the left from the fifth on. So Lemm 1 ivies ever element in the top five ros of the entire rr n hene in the entire rr n n etension to the right if n. Thus e f n g. Similrl e ivies ll oeffiients in the tringle of lest support ontining 2 n. Hene s for e ivies ever element in the rr n e e f n e g. Sine similr rgument hols for oth f n g it follos tht = e = f = g s lime. Oserve tht s note in the proof of Theorem 2 the rr in Figure n e etene ritrril fr to the right sujet to the onition tht n element of eh of S S S n S pper mong the top four elements in eh sueeing igonl sloping onr n to the left. Moreover eept for n the elements in the ottom five ros of the tringle n e ompletel ritrril hosen from mong S S S n S ithout ffeting the vliit of the result. Of ourse similr sitution previls for ll other vlues of n s ell.. Aknolegement The uthors re grteful to the referee to severl helpful omments. Referenes [1] S. Ano n D. Sto A neessr n suffiient onition tht rs of str onfigurtion in Psl s tringle over its enter ith respet to GCD n LCM Applitions of Fioni Numers E. G. E. Bergum et l. Kluer Dorreht (1) 11. [2] B. Goron D. Sto n E. Struss Binomil oeffiients here prouts re perfet kth poers Pifi Journl of Mthemtis 11.2 (1) 00. [] H. Goul A ne gretest ommon ivisor propert of the inomil oeffiients The Fioni Qurterl 10. (12) 2. [] A. Hillmn n V. Hoggtt Jr. A proof of Goul s Psl hegon onjeture The Fioni Qurterl 10. (12). [] V. Hoggtt Jr. n W. Hnsell The hien hegon squres The Fioni Qurterl.2 (11) [] E. Korntve Etensions of the GCD str of Dvi theorem The Fioni Qurterl 2.2 (1) [] C. Long Arrs of inomil oeffiients hose prouts re squres The Fioni Qurterl 11. (1). [] C. Long n V. Hoggtt Jr. Sets of inomil oeffiients ith equl prouts The Fioni Qurterl 12.1 (1) 1. [] C. Long W. Shul n S. Ano An etension of the GCD str of Dvi theorem The Fioni Qurterl. (200) [10] C. Moore More hien hegon squres The Fioni Qurterl 11. (1) 2. [11] Z. Usiskin Perfet squre ptterns in Psl s tringle Mthemtis Mgine. (1) NOVEMBER

5 THE FIBONACCI QUARTERLY MSC2010: 11B 11A0 Deprtment of Pure n Applie Mthemtis Wshington Stte Universit Pullmn WA 1 E-mil ress: tl102@verion.net Deprtment of Mthemtis n Computer Siene Northest Nrene Universit Nmp ID E-mil ress: ekorntve@nnu.eu 1 VOLUME NUMBER