Squares from D( 4) and D(20) Triples

Transcription

1 dvances in Pure Mathematics - doi:/apm Published Online September ( Squares from D( ) D() Triples bstract Zvono Čerin Koperniova 7 Zagreb roatia cerin@mathhr Received May ; revised May ; accepted June We study the eight infinite sequences triples natural numbers Fn Fn Fn7 B F n Fn Fn 7 Fn Fn F n D Fn Fn F n L n L n Ln7 L L L L L L L L L n n n 7 n n n n n n The sequences B D are built from the Fibonacci numbers F n while the sequences from the Lucas numbers L n Each triple in the sequences B D has the property D (i e adding to the product any two different components them is a square) Similarly each triple in the sequences has the property D We show some interesting properties these sequences that give various methods how to get squares from them Keywords: Fibonacci Numbers Lucas Numbers Square Symmetric Sum lternating Sum Product omponent Introduction For integers a b c let us write a c provided a b= c For the triples X = abc Y = d e f X = abc the notation Y X X means that d e bc a ca b f ab c When Y = let us write X X Y for X X Hence X is the D triple (see [] ) if only if there is a triple X such that X X We now construct the infinite sequences B D the D -triples the D -triples They are F F n n F B n 7 F n Fn Fn7 F n F n Fn D F n Fn Fn L n Ln Ln7 LnLnLn7 L n Ln Ln LnL nln where the Fibonacci Lucas sequences natural numbers F n Ln are defined by the recurrence relations F = F = Fn = F n Fn for n L = L = Ln = Ln Ln for n The numbers F mae the integer sequence from [] while the numbers L mae For an integer let us use π p r for F n L n F n b L n The goal this article is to explore the properties the sequences B D Each member these sequences is an Euler D - or D -triple (see []) so that many their properties follow from the properties the general (pencils ) Euler triples (see [] ) It is therefore interesting to loo for those properties in which at least two the sequences appear This paper presents several results this ind giving many squares from the components various sums products the sequences B D Most our theorems have also versions for the associated sequences = π π π B = π π π = π D = π = = π π = π π that satisfy the relations B B D D The overall principle in this paper is that if you can get complete squares by adding a fixed number to the products different components some triples natural numbers then you will be able to get complete squares by adding some other fixed numbers to all inds expressions constructions built from the components these triples Our tas was to find out these numbers to identify those expressions constructions ll results in this paper are identities among Fibonacci opyright SciRes

2 Z ČERIN 7 /or Lucas numbers varied difficulty We shall write down the pros only a small portion them to save the space leaving the rest to the dedicated reader In most cases we prove or only outline the pro the first among several parts the theorem The other parts have similar pros sometimes with far more complicated details Following this introduction in the section we first show that the selected products four components among triples from either the sequences B D or the sequences B D become squares by adding some fixed integers The Section considers the various products two symmetric quadratic sums components sees to get squares in the same way (by adding a fixed integer) The next Section does a similar tas for certain products four symmetric linear sums components In the Section the numerous products two sums squares components are shown as differences squares The long Section contains similar results for products two symmetric linear sums components the three natural products (dot forward shifted dot bacward shifted dot) two triples integers Finally the last section 7 replaces these dot products with the two forms a stard vector product in the Euclidean -space Squares from Products omponents The relations imply that the components satisfy Our first theorem shows that the product is in a similar relation with respect to Of course the other products as well as BB etc exhibit a similar property The missing cases from the list coincide with the one the previous cases Theorem The following hold for the products components: p BB p p p p DD p BB p p Pro Let = = = Since j j F j = L = j j j it follows that n n n7 n7 = = n n = = n 7 n7 fter the substitutions = = n the sum the product becomes However this is precisely the square p This shows the first relation The version the previous theorem for the sequences B D is the following result Notice that in this theorem there are no repetitions cases Theorem The products the components satisfy: p BB p7 r DD r p7 BB p p DD p p BB p r DD p n n Pro Since = n n = n n = n n = the sum after the substitutions = = n becomes However the square p has the same value This proves the first relation p The same ind relations hold also for the products components from four among the sequences B D Theorem The relations that hold for the products components: B D p p p opyright SciRes

3 Z ČERIN B p BD p D p B p Dp D p B D p p p Pro Since B = = the first relation is the consequence the first relation in Theorem Similarly the fifth relation follows from the sixth relation in the Theorem The other relations in this theorem have pros similar to the pros Theorems There is again the version the previous theorem for the products components from four among the sequences B D Theorem The products components satisfy: B p7 r D p D p B p r B r D r B p 7 p D r BD r D p Pro The first the sixth the ninth the tenth relations are the easy consequences the second the last the seventh the fourth relations in Theorem In order to prove the second relation note that the components are n n n n n n n n It is now clear from the pro Theorem that the sum is precisely the square r This requires the identities π = p π = p pp = r Squares from Symmetric Sums Let : be the basic symmetric functions defined for x = abc by x a b c x bc ca ab x abc Let : be defined by = x bc ca ab x = a b c Note that x is the determinant the matrix abc (see [] ) For the sums the components the following relations are true Theorem The following is true for the sums the components: p p B r p 7 7 7r p D p 7 p p B p r D p Pro Since B = r = r the sum B is r that we recog- nize as the square p This proves the sixth relation B p The sums B have constant values On the other h π r D r Theorem The following is true for the sums the components: 7p p B p 7p 7 p p D p p = π π 7 Pro Since = π π the sum is the square p p This proves the third relation Some similar relations mae up the following two results Theorem 7 The following is true for the sums the components: B p B r D r B Pro Since = π = it follows that = = B opyright SciRes

4 Z ČERIN p B = π = Theorem The following is true for the sums the components: B r Pro Since = r r B = r r B = = it follows that = r r = r Products Sums as Differences Squares The products the sums the components the four triples among B D show the same ind relations This is also true for the associated triples B D Notice that in the next four theorems the added third number is always a square so that the product on the left h side in each relation is a difference squares Theorem The following relations hold for the sums : B p p r D p + p B r + p B D p p D p B p p p D p p7 B p 7 p B D p D p p7 Pro The sums the components B are equal π π π π Hence the sum B is the square p This proves the above first relation Theorem The following relations hold for the sums : B p p D r7 D r B p r p p 7 D p p B p r B D p p D p B are equal π π Pro The sums the components Hence the product B is the square p since π = p This proves the above first relation In the next result we combine the sums in each product Theorem The following relations hold for the sums : B p r B p p 7 r p r p D p p D p 7 p B r p B D p p D r r D p p Pro With the above information about the sums the components B the sum B is r p = p r p π π = p p This proves the first relation Theorem The following relations hold for the sums : B p r B p p p p p D p7 p D p p7 opyright SciRes

5 Z ČERIN B 7 p B p p7 B D p r B D p r 7 7 D D p r p p Pro The sums the components are equal π π π Hence the sum the product B is p π π = pp p p= p r This proves the above first relation Squares from the Sums Squares B For a natural number > let the sums : powers be defined for x = abc by x = a b c x = a b c We proceed with the version the Theorem for the sums the squares components Theorem The following relations are true for the sums : p r B 7 p p 7r p D 7 r p p r B 7p p 7 7 7r p D 7r p Pro Since are 7 p p 7p p the difference is equal where are 77 7 But one can easily chec that this is the square p r This concludes the pro the first relation The next is the version the Theorem for the alternating sums the squares components Theorem The following relations are true for the sums : p r B p r r p D r r 7 r p B 7r p 7 r p D p p Pro Notice that the alternating sums squares components are r r r r Hence the sum is equal However one can easily chec that this is the square p r This proves the first relation Multiplied by five these products the sums components show the same behavior Theorem For the sums the following relations hold: p p B pp p p D p p p r B r p 7 r p D r r Pro With the above values the sum 7 is equal But this is the square p p This outlines the pro the first relation Squares from the Products Let us introduce three binary operations on the set triples integers by the rules abc uvw = aubvcw opyright SciRes

6 Z ČERIN abc uvw = avbwcu the operations are also the source abc uvw = awbucv squares from components the sixteen sequences This section contains four theorems which show that Theorem The following relations hold for the sequences : B p p p 7 D B 7 r p r p 7 B D r D p 7 B p r r r p p D B B D D r r 7 p p7 r B p D B r r 7p p BD p r D p 7 B is the square p Pro Since B = 7p p This proves the first relation =7p p it follows that the difference Theorem 7 The following relations hold for the sequences : ( B ) ( ) p ( ) ( ) p r D B p r p p 7 B D p r D p p 7p p r p B r r r r D B r r p B D D opyright SciRes

7 Z ČERIN p 7 p p B 77 D B p p p p p B D p D Pro Since the sums B are r r it follows that the sum B is B r r p i e the square p This concludes the pro the first relation Theorem The following relations hold for the sequences : r p p r 7 p r p p D B B D D 7 p p p B p p p 7 D B p p r p B D p r D p r p B p 7 D p B p p 7 r p B D p D Pro Since the sums B are π π π it fol- B is lows that the sum p π π i e B which is the square r p This is the outline the pro the first relation Theorem The following relations hold for the sequences : p r p p 7 p r p p D B p 7 p r r B D D 7 opyright SciRes

8 p Z ČERIN B p r 7 D 7 B p p p p B D p p D p p p B p p r p D B p B D p D Pro Since the sums B are 7 π π 7π it follows that the sum B is π 7π 7 p ie 7 7 which is the square 7 p r This is the outline the pro the first relation 7 Squares from the Products This section uses the binary operations defined by abc de f = bfcecd af aebd abc de f= bfcecd af aebd Note that restricted on the stard Euclidean -space the product is the familiar vector cross-product Theorem The following relations hold for the triples : Pro Since the sums r B B p r D D r are π π it follows that the sum is pp ie the square r This concludes the pro the first relation Theorem The following relations hold for the triples : 7 p p B B p 7 7 r r ( ) D D p p Pro Since the sums are r r 7 r r 7 it follows that the difference the product 7 is opyright SciRes

9 Z ČERIN r r which simplifies to 7 7 ie to the square p p This proves the first relation Notice that in our final result the third added constant value is in all cases the complete square Theorem The following relations hold for the triples : B p p p 7r p p p D B B D r p D p p r B p 7 D r p B r p 7 B D r p D p B Pro Since the sums are p r p p it follows that the sum B is the product p r p p which simplifies to the square p This proves the first relation References [] E Brown Sets in Which xy Is lways a Square Mathematics omputation Vol No 7 pp - [] N Sloane On-Line Encyclopedia Integer Sequences njas/sequences/ [] L Euler ommentationes rithmeticae I Opera Omnia Series I volume II BG Teubner Basel [] Z Čerin On Pencils Euler Triples I (in press) [] Z Čerin On Pencils Euler Triples II (in press) [] M Radić Definition Determinant Rectangular Matrix Glasni Matematic Vol No pp 7- opyright SciRes