SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES

Transcription

1 #A17 INTEGERS ), SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA dmk5075@psu.edu James A. Selles Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA sellesj@math.psu.edu Robet G. Vay Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA gv106@psu.edu Received: 5/30/08, Accepted: 3/4/09 Abstact Recetly, Lovejoy itoduced the costuct of ovepatitio pais which ae a atual geealizatio of ovepatitios. Hee we geealize that idea to ovepatitio k- tuples ad pove seveal cogueces elated to them. We deote the umbe of ovepatitio k-tuples of a positive itege by p k ) ad pove, fo example, that fo all 0, p t 1 t + ) 0 mod t) whee t is pime ad is a quadatic oesidue mod t. 1. Itoductio As defied by Coteel ad Lovejoy [5], a ovepatitio of a positive itege is a o-iceasig sequece of atual umbes whose sum is i which the fist occuece of a pat may be ovelied. Fo example, the ovepatitios of the itege 3 ae 3, 3, 2 + 1, 2 + 1, 2 + 1, 2 + 1, , The umbe of ovepatitios of a positive itege is deoted by p), with p0) 1 by defiitio. Thus p3) 8 fom the above example. As oted i Coteel ad Lovejoy [5], the geeatig fuctio fo ovepatitios is p)q 1 + q 1 q. 0 As the topic of ovepatitios has aleady bee examied athe thooughly [3, 4, 5, 6, 7, 8, 10, 11], we look to ew costuctios. Oe such costuctio is that of a ovepatitio pai of a positive itege, defied by Lovejoy [9] as a pai of ovepatitios wheei the sum of all listed pats is. Fo example, the ovepatitio pais of 2 ae 2 ; ), 2 ; ), ; 2), ; 2), ; ), ; ), ; 1 + 1), ; 1 + 1), 1 ; 1), 1 ; 1), 1 ; 1), 1 ; 1).

2 INTEGERS: ) 182 Lovejoy deoted the umbe of ovepatitio pais of a positive itege by pp), with pp0) 1 by defiitio. Thus pp2) 12 fom the above example. Followig lies simila to that fo ovepatitios, the geeatig fuctio fo ovepatitio pais is ) 1 + q pp)q 2 1 q. 0 Seveal aithmetic popeties of both ovepatitios ad thei pais have appeaed i the liteatue. Sice ou iteest hee is pimaily o coguece popeties, thee ae a few theoems that ae especially otewothy. The fist oe is staightfowad ad pove ituitively. Theoem 1. Fo all > 0, p) 0 mod 2). Next we have a theoem easily pove usig esults of Mahlbug [10]. Theoem 2. Fo all > 0, p) { 2 mod 4) if is a squae, 0 mod 4) othewise. Seveal othe cogueces i aithmetic pogessios wee pove by Hischho ad Selles. Fo example, the followig wee pove i [7]. Theoem 3. Fo all 0, p5 + 2) 0 mod 4), p5 + 3) 0 mod 4), p4 + 3) 0 mod 8), ad p8 + 7) 0 mod 64). Also, Hischho ad Selles [6] poved that p) satisfies cogueces modulo o-powes of 2 by povig the followig: Theoem 4. Fo all 0 ad all α 0, p 9 α )) 0 mod 12). Fially, we ote a theoem pove by Bigma ad Lovejoy [2]. This esult povides much ispiatio fo the mai esult i the ext sectio. Theoem 5. Fo all 0, pp3 + 2) 0 mod 3). We ow itoduce a geealizatio of ovepatitio pais. A ovepatitio k-tuple of a positive itege is a k-tuple of ovepatitios wheei all listed pats sum to. We deote the umbe of ovepatitio k-tuples of by p k ), with p k 0) 1 by

3 INTEGERS: ) 183 defiitio. Cosequetly, the umbe of ovepatitio pais of is deoted as p 2 ). The geeatig fuctio fo p k ) is easily see to be p k )q 0 ) 1 + q k 1 q. The aim of this ote is to pove seveal coguece popeties fo families of ovepatitio k-tuples. I the pocess, we will pove seveal atual geealizatios of esults quoted above. 2. Results fo Ovepatitio k-tuples Ou fist theoem of this sectio povides a atual geealizatio of Bigma ad Lovejoy s Theoem 5 above. Moeove, the poof techique is extemely elemetay, makig this a vey satisfyig esult. Theoem 6. Fo all 0, p t 1 t + ) 0 mod t), whee t is a odd pime ad is a quadatic oesidue mod t. Remaks. Fist, ote that the t 3 case of this theoem is exactly Theoem 5. Secodly, ote that, fo each odd pime t, this theoem povides t 1 2 coguece popeties fo p t 1 ). Poof. Coside the followig geeatig fuctio maipulatios: ) 1 + q p t 1 )q i t 1 0 [ ] t q i [ ] t [ 1 + q i ] 1 + q i [ ] [ 1 + q ti ] 1 q ti 1 + q i [ ] pm)q tm 1 + q i m0 pm)q tm m0 s mod t) sice t is pime 1) s q s2 thaks to Gauss [1, Co. 2.10].

4 INTEGERS: ) 184 But ote that t+ ca eve be epeseted as tm+s 2 fo some iteges m ad s if is a quadatic oesidue mod t. This implies that p t 1 t + ) 0 mod t) fo all 0. The ext theoem is a boad geealizatio of Theoem 1. It is foud with poof i [12], but is icluded hee fo the sake of completeess. We equie a bief techical lemma. Lemma 7. Let m be a oegative itege. Fo all 1 2 m, ) 2 m 2 0 mod 2 m+1 ). Poof. Let od 2 N) be the expoet of the highest powe of 2 dividig N. Thus, fo example, od 2 8) 3 while od 2 80) 4. To pove Lemma 7, we eed to pove that ) 2 m od 2 )2 m ) Note that ) 2 m 2 od 2 )2 m 2 m 1)2 m 2) 2 m ) 1)) od 2 2! ) 2 m+ od 2! m + od 2!) ) m whee x is the floo fuctio of x. Now assume c c c t 2 t whee each c i {0, 1}. The c c c t 2 t c c c t 2 t 2 + c c c t 2 t 3 + c t )c )c )c t 1)c t c 0 + c 1 + c c t ) 1

5 INTEGERS: ) 185 sice at least oe of the c i must equal 1. Theefoe, ) 2 m od 2 )2 ) m m + 1) m + 1. This is the desied esult as oted i 1) above. We ae ow i a positio to pove the followig theoem: Theoem 8. Let k2 m ), whee m is a oegative itege ad is odd. The, fo all positive iteges, we have p k ) 0 mod 2 m+1 ). Poof. [ 1 + q p k )q i 0 [ 1 + q i ] k ] 2 m ) [ ] 1 + q i 2 m) [ 1 + ] 2 m) [1 + 2qi 2 m 2 m q )2 i ) ]) 1 mod 2 m+1 ) by Lemma 7. The followig theoem is ispied by Theoem 2. As with Theoem 8, it pimaily higes upo the use of the biomial theoem. Theoem 9. Let k2 m ), m > 0 ad is odd. The, fo all 1, { 2 m+1 mod 2 m+2 ) if is a squae o twice a squae, p k ) 0 mod 2 m+2 ) othewise.

6 INTEGERS: ) 186 Poof. We pove this esult by iductio o m. Basis Step. Let m 1. We must show that { 4 mod 8) if is a squae o twice a squae, p 2 ) 0 mod 8) othewise. ) 1 + q p 2 )q i 2 0 [ ] 2 p)q p)q + p)q 2 squae p)q + p)q squae + 2 p)q + 2 squae squae p)q + p)q 2 2 p)q Fom Theoem 2, we kow that p) 2 o 6 mod 8) whe is a squae ad p) 0 o 4 mod 8) othewise. Sice 2 0, 2 4, 6 0, 6 4, 0 0, 0 4, ad 4 4 ae all coguet to 0 mod 8), 2 p)q squae ad 2 p)q 0 mod 8), p)q 0 mod 8), 2 p)q 0 mod 8).

7 INTEGERS: ) 187 This gives ) ) 2 p 2 )q p 2 )q 2 + p 2 )q 2 0 ) ) q q 2 mod 8) mod 8) agai thaks to Theoem 2. Give that q 1 + q 2 + ) 2 q 21 + q 22 + ) + 2 q ), we the have ) p 2 )q q q m1 [ )] q m2 + q 22 mod 8) q , 2>0 1 2 ) ) j 4 j q m2 + q 22 j j0 m1 ) q m2 + q 22 mod 8) sice is odd. m1 mod 8) This poves the esult eeded fo the basis step. Iductio Step. Assume that { 2 m+1 mod 2 m+2 ) if is a squae o twice a squae, p 2m )) 0 mod 2 m+2 ) othewise. We must show that p 2 m+1 )) { 2 m+2 mod 2 m+3 ) if is a squae o twice a squae, 0 mod 2 m+3 ) othewise. Coside the geeatig fuctio fo p 2 m+1):

8 INTEGERS: ) q p 2 m+1 ))q i )2 m+1 ) 0 ) ) 1 + q i 2 m 2 ) 2 p 2m ))q ot sq. ad ot p 2m ))q + squae o p 2m ))q ) 2 ) ) p 2m ))q + p 2m ))q + 2 squae o squae o ) p 2m ))q squae o ot sq. ad ot p 2m ))q ) ) ) p 2m ))q + p 2m ))q. ot sq. ad ot ot sq. ad ot Usig a vey simila agumet about the coefficiets to that of the basis step, we use the iductio hypothesis to coclude that p 2 m+1 ))q p 2m ) 2 )q 2 + p 2m )2s 2 )q 2s2 0 s1 ) 2 + p 2m ) 2 )q 2 + p 2m )2s 2 )q 2s2 mod 2 m+3 ) s p 2m ) 2 )q 2 + p 2m )2s 2 )q )mod 2s2 2 m+3 ). s1

9 INTEGERS: ) 189 We kow that all coefficiets of the last tem ae coguet to 2 m+1 o 2 m m+2 mod 2 m+3 ) fom the iductio hypothesis. But the last tem is multiplied by 2. So the all coefficiets ae coguet to 2 m+2 mod 2 m+3 ) o 2 m m+3 2 m+2 mod 2 m+3 ), which implies ) p 2 m+1 ))q m+2 q 2 + q 2s2 mod 2 m+3 ). 0 s1 This completes the iductio ad poves the theoem. Ackowledgmets We thak the Pe State Cete fo Udegaduate Reseach i Mathematics PSU CURM) fo bigig the authos togethe. It is also impotat to ote the Roald E. McNai Post-Baccalaueate Achievemet Pogam s ivolvemet i allowig Robet to coduct eseach ude the supevisio of D. Selles duig its 2008 summe pogam. Refeeces [1] G. E. Adews, The Theoy of Patitios, Cambidge Uivesity Pess 1984). [2] K. Bigma ad J. Lovejoy, Rak ad cogueces fo ovepatitio pais, It. J. of Numbe Theoy ), [3] S. Coteel, W. Goh, ad P. Hitczeko, A local limit theoem i the theoy of ovepatitios, Algoithmica 46, o ), [4] S. Coteel ad P. Hitczeko, Multiplicity ad umbe of pats i ovepatitios, A. Comb ), [5] S. Coteel ad J. Lovejoy, Ovepatitios, Tas. Ame. Math. Soc ), [6] M. D. Hischho ad J. A. Selles, A ifiite family of ovepatitio cogueces modulo 12, INTEGERS ), Aticle A20. [7] M. D. Hischho ad J. A. Selles, Aithmetic elatios fo ovepatitios, J. Combi. Math. Combi. Comput ), [8] J. Lovejoy, Ovepatitios ad eal quadatic fields, J. Numbe Theoy ), [9] J. Lovejoy, Ovepatitio pais, A. Ist. Fouie ), [10] K. Mahlbug, The ovepatitio fuctio modulo small powes of 2, Discete Math ), o. 3, [11] Ø. Rødseth ad J. A. Selles, O m-ay ovepatitios, A. Comb ),

10 INTEGERS: ) 190 [12] R. Vay, Some ovepatitio k-tuple cogueces, The Pe State McNai Joual ), to appea.