New index matrix representations of operations over natural numbers

Transcription

1 Note on Number Theory and Dicrete Mathematic Print ISSN Online ISSN Vol No DOI: 07546/nntdm New index matrix repreentation of operation over natural number Lilija Atanaova Intitute of Information and Communication Technologie Bulgarian Academy of Science 2 Acad Georgi Bonchev Str Sofia 3 Bulgaria lcatanaova@gmailcom Received: 8 July 207 Accepted: 3 January 208 Abtract: Two new operation over index matrice are introduced Their poible application in number theory i dicued and illutrated with example related to the canonical repreentation of the natural number and with two extended Fibonacci equence Keyword: Fibonacci equence Index matrix Natural number 200 Mathematic Subject Claification: B39 Introduction The idea of the concept of an Index Matrix (IM) wa dicued for the firt time in [] and introduced formally in [4] There the firt operation over IM were given The baic reult related to IM were included in [5] In thi book a example IM-repreentation of ome operation in number theory were decribed In the preent paper extenion of ome operation dicued in [5] are given and new example are decribed 2 Preliminarie Following [5] we define the concept of an IM and ome operation over them Let I be a fixed et of indice and R be the et of real number Let operation : R R R be fixed For example they can be { + max min} or other 53

2 Let the tandard et K and L atify the condition: K L I Let over thee et the tandard et-theoretical operation be defined We call IM with real number element (R-IM) the object: [K L {a ki l j }] k a k l a k l 2 a k l n k 2 a k2 l a k2 l 2 a k2 l n where k m a kml a kml 2 a kml n K = {k k 2 k m } and L = {l l 2 l n } and for i m and for j n : a ki l j R Let the IM A be given and let K and L be two indice Now following [7] and [5] we introduce the following four aggregation operation over it: Max-row-aggregation ρ max (A ) = Min-row-aggregation max i m a k i l max a k i l 2 max a k i l n i m i m ρ min (A ) = Sum-row-aggregation min i m a k i l min i m a k i l 2 min i m a k i l n ρ um (A ) = Average-row-aggregation m m m a ki l 2 a ki l n a ki l ρ ave (A ) = m m m m a ki l a m ki l 2 a m ki l n 3 Main reult A it wa mentioned in [5] it i well-known (ee eg [9 0]) that each natural number m ha a canonical repreentation m = k p α i i where k α α 2 α k are natural number and p p 2 p k are different prime number Let u alway uppoe that p < p 2 < < p k Thi condition i only for convinience becaue there i no pecific an order of the row and column in an IM but thee are labeled by indice 54

3 Then a it i hown in [5] the natural number m ha the following IM-interpretation: IM(m a) = p p 2 p k a α α 2 α k where a i an arbitrary ymbol in a particular cae the ame m In thi cae for brevity we write IM(m m) = IM(m) In [2] the function et i introduced for the above number m by et(m) = {p p k } Firt we generalize the example from [5] Let u have natural number N N 2 N and let et(n i ) = {p p k } Therefore for each i ( i ) : N i = k contruct the IM j= p α ij j where α ij 0 and k α ij Now we j= p p 2 p k IM(N N ) = N α α 2 α k N α α 2 α k For example if N = 2 N 2 = 27 N 3 = 30 N 4 = 50 then thee number have the canonical repreentation N = N 2 = 3 3 N 3 = N 4 = and IMrepreentation N 2 0 IM(N N 2 N 3 N 4 ) = N N 3 N 4 2 The reult of application of the aggregation operation over IM IM(N N ) will be repectively: ρ max (IM(N N ) ) = ρ min (IM(N N ) ) = ρ um (IM(N N ) ) = p p 2 p k max i i max i2 i max ik i m p p 2 p k min i i min i2 i min ik i m p p 2 p k α i2 α ik α i i i i m ρ ave (IM(N N ) ) = Now we ee immediately that: p p 2 p k α i α i2 i i i m α ik 55

4 IM ρ max (IM(N N ) ) repreent the leat common multiple of the number N N ; IM ρ min (IM(N N ) ) repreent the greatet common divior of the number N N ; IM ρ um (IM(N N ) ) repreent the product of the number N N ; IM ρ ave (IM(N N ) ) repreent the geometric average of the number N N It i worth mentioning that the fourth cae i not dicued in [5] For the above example thee formula obtain the following form: ρ max (IM(N N 4 ) ) = p p 2 p ρ min (IM(N N 4 ) ) = ρ um (IM(N N 4 ) ) = p p 2 p p p 2 p p p 2 p 3 ρ ave (IM(N N 4 ) ) = The fourth cae give the idea for introducing of the following new aggregation operation: p p 2 p k ρ geo (IM(N N ) ) = α i i i α i2 i m α ik For the above example we obtain: ρ geo (IM(N N 4 ) ) = p p 2 p but we mut mention immediately that the element of the newly contructed IM do not correpond to geometric average of N N They do not correpond to any known arithmetic operation Second we introduce two new IM-operation in which the indice when they are real (natural) number participate with additional role Let u have the IM A = [K L {a ki l j }] k a k l a k l 2 a k l n k 2 a k2 l a k2 l 2 a k2 l n k m a kml a kml2 a kmln 56

5 where K = {k k 2 k m } R and L = {l l 2 l n } R and for i m and for j n : a ki l j R Now we define A = k a k l l a k l 2 l 2 a k l n l n k 2 a k2 l l a k2 l 2 l 2 a k2 l n l n k m a kml l a kml 2 l 2 a kml n l n and A = k a k l k a k l 2 k a k l n k k 2 a k2 l k 2 a k2 l 2 k 2 a k2 l n k 2 k m a kml k m a kml 2 k m a kml n k m For example for the IM IM(N N ) we obtain p p 2 p k IM(N N ) = N α p α 2 p 2 α k p k N α p α 2 p 2 α k p k In [5] the following average operation i defined over IM A: σ um (A ) = k k m n a k l j j= n a kmlj j= Now for our example we obtain σ um ( IM(N N ) ) = N N k a j p j j= k a j p j j= In [3] function ζ i defined over the natural number m from Section a follow: ζ(m) = k α i p i 57

6 Now for our example we obtain σ um ( IM(N N ) ) = N ζ(n ) N ζ(n ) We finih with another example related to Fibonacci equence In [6] the following extenion of the Fibonacci equence call 2-Fibonacci equence wa introduced a follow: The firt ten term of thi equence are: α 0 = a β 0 = b α = c β = d α n+2 = β n+ + β n n 0 β n+2 = α n+ + α n n 0 n α n β n 0 a b c d 2 b + d a + c 3 a + c + d b + c + d 4 a + b + 2c + d a + b + c + 2d 5 a + 2b + 2c + 3d 2a + b + 3c + 2d 6 3a + 2b + 4c + 4d 2a + 3b + 4c + 4d 7 4a + 4b + 7c + 6d 4a + 4b + 6c + 7d 8 6a + 7b + 0c + d 7a + 6b + c + 0d 9 a + 0b + 7c + 7d 0a + b + 7c + 7d Now we can contruct the following two IM correponding repectively to the member of equence {α n } n 0 and {β n } n 0 eg for n 9: IM({α n } 0 n 9 ) = a b c d α α α α 3 0 α 4 2 α α α α α

7 We ee again that IM({β n } 0 n 9 ) = a b c d β β β β 3 0 β 4 2 β β β β β and σ um ( IM({α n } 0 n 9 ) = σ um ( IM({β n } 0 n 9 ) = α 0 a α c α 2 b + d α 3 b + c + d α 4 a + b + c + 2d α 5 a + 2b + 2c + 3d α 6 3a + 2b + 4c + 4d α 7 4a + 4b + 7c + 6d α 8 6a + 7b + 0c + d α 9 a + 0b + 7c + 7d α 0 b α d α 2 a + c α 3 a + c + d α 4 a + b + 2c + d α 5 2a + b + 3c + 2d α 6 2a + 3b + 4c + 4d α 7 4a + 4b + 6c + 7d α 8 7a + 6b + c + 0d α 9 0a + b + 7c + 7d 4 Concluion The apparatu of index matrice ha already found ome application in the area of number theory (eg in [5 8] and other) but it i clear that thee publication are only the firt tep in thi 59

8 direction of reearch On one ide the new operation can find application in a lot of other area and on the other ide the above reearch can be perceived a the firt tep in applying the new operator to element of different equence which i an object of further reearch in the future Reference [] Atanaov K (984) Condition in Generalized net Proc of the XIII Spring Conf of the Union of Bulg Math Sunny Beach April [2] Atanaov K (984) On one problem of A Mullin Bulletin of Number Theory and Related Topic VIII 3 5 [3] Atanaov K (987) New integer function related to ϕ and σ function Bulletin of Number Theory and Related Topic XI 3 26 [4] Atanaov K (987) Generalized index matrice Compt Rend de l Academie Bulgare de Science [5] Atanaov K (204) Index Matrice: Toward an Augmented Matrix Calculu Springer Cham [6] Atanaov K Atanaova L & Saelov D (985) A new perpective to the generalization of the Fibonacci equence Fibonacci Quarterly [7] Atanaov K Sotirova E & Bureva V (203) On index matrice Part 4: New operation over index matrice Advanced Studie in Contemporary Mathematic [8] Leyendekker J V Shannon A G & Rybak J (2007) Pattern recognition: Modular Ring & Integer Structure Raffle KvB Monograph No 9 North Sydney [9] Mitrinović D S Sándor J & Crtici B (995) Handbook of Number Theory Kluwer Dordrecht [0] Nagell T (950) Introduction to Number Theory John Wiley & Son New York 60