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

We study the eight infinite sequences of triples of natural numbers   2 1 2 3 2 7 ,4 , n n n A F F F     ,  2 1, n B F    2 5 2 7 4 , n n F F   ,   2 1 2 1 2 3 ,5 ,4     n n n C F F F ,   2 3 2 1 ,4 ,5    n n D F F 2 3  n F and   2 1 2 3 , 2 7 , n n n L    ,4 L L     2 1,4 L L   2 5 2 7 n n n    , , L   2 1 2 1 , 2 3  n L ,5    n n L L  ,4   2 3 2 1 2 3 . The sequences and ,4  L L  ,5    n n n L , , A B C D are built from the Fibonacci numbers n F while the sequences , , and from the Lucas numbers n . Each triple in the sequences     L , , A B C and D has the property (i. e., adding   4 D  4  to the product of any two different components of them is a square). Similarly, each triple in the sequences , , and  has the property . We show some interesting properties of these sequences that give various methods how to get squares from them.    20 D 

, 4 , 4 , ,5 ,4 ,4 ,5 ,    [1] ) if and only if there is a triple X  such that k X X   .We now construct the infinite sequences A , , and of the D riples and  , and  the   D iples.They are , , and , 4 , where the Fibonacci and Lucas sequences of natural numbers and n are defined by the recurrence relations , , and , , for .
The numbers k L L  F make the integer sequence from [2] while the numbers make .For an integer , let us use , and for , , F n k  and .

4
L n k


The goal of this article is to explore the properties of the sequences A , , member of these sequences is an Euler -triple (see [3]) so that many of their properties follow from the properties of the general (pencils of) Euler triples (see [4,5] ).It is therefore interesting to look for those properties in which at least two of the sequences appear.This paper presents several results of this kind giving many squares from the components, various sums and products of the sequences A , , , Most of our theorems have also versions for the associated sequences The overall principle in this paper is that if you can get complete squares by adding a fixed number to the products of different components of some triples of natural numbers then you will be able to get complete squares by adding some other fixed numbers to all kinds of expressions and constructions built from the components of these triples.Our task was to find out these numbers and to identify those expressions and constructions.
All results in this paper are identities among Fibonacci and/or Lucas numbers of varied difficulty.We shall write down the proofs of only a small portion of them to save the space leaving the rest to the dedicated reader.In most cases we prove or only outline the proof of the first among several parts of the theorem.The other parts have similar proofs sometimes with far more complicated details.
Following this introduction, in the section 2, we first show that the selected products of four components among triples from either the sequences A , , , ,  , , and or the sequences , and become squares by adding some fixed integers.
The Section 3 considers the various products of two symmetric quadratic sums of components and seeks to get squares in the same way (by adding a fixed integer).
The next Section 4 does a similar task for certain products of four symmetric linear sums of components.
In the Section 5 the numerous products of two sums of squares of components are shown as differences of squares.
The long Section 6 contains similar results for products of two symmetric linear sums of components of the three natural products (dot, forward shifted dot and backward shifted dot) of two triples of integers.
Finally, the last section 7 replaces these dot products with the two forms of a standard vector product in the Euclidean 3-space.

Squares from Products of Components
. Our first theorem shows that the product 20 A A   is in a similar relation with respect to 9.
Of course, the other products 3 1 3 1 , 1 2 1 2 as well as 2 3 2 3 , etc. exhibit a similar property.The missing cases from the list coincide with the one of the previous cases.
The following hold for the products of components: After the substitutions The products of the components of  A , , satisfy: A A   and 1, after the substitutions However, the square of has the same value.This proves the first relation 6 The same kind of relations hold also for the products of components from four among the sequences A , , , ,  , , and .

B C D
   Theorem 3. The relations that hold for the products of components: Proof.Since 3 3 and 3 3 , the first relation is the consequence of the first relation in Theorem 1.Similarly, the fifth relation follows from the sixth relation in the Theorem 1.
The other relations in this theorem have proofs similar to the proofs of Theorems 1 and 2.
There is again the version of the previous theorem for the products of components from four among the sequences A  , , , Proof.The first, the sixth, the ninth and the tenth relations are the easy consequences of the second, the last, the seventh and the fourth relations in Theorem 2.
In order to prove the second relation, note that the components 2  A , , and are It is now clear from the proof of Theorem 1 that the sum of 2 3 2 3 and 16 is precisely the square of 5 .This requires the identities p p

Squares from Symmetric Sums
Let be the basic symmetric functions defined for , , : , , a b c (see [6] ).
For the sums 2 Theorem 8.The following is true for the sums 2  of the components:

Products of Sums as Differences of Squares
The products of the sums 1  and * 1  of the components of the four triples among A , , , , , , and show the same kind of relations.This is also true for the associated triples Notice that in the next four theorems the added third number is always a square so that the product on the left hand side in each relation is a difference of squares.
Theorem 9.The following relations hold for the sums Proof.The sums of the components Proof.The sums of the components Hence, the product  p .This proves the above first relation.
In the next result we combine the sums 1  and * 1  in each product.
Theorem 11.The following relations hold for the sums 1  and * 1  : 16 16 Proof.With the above information about the sums of the components Proof.The sums of the components and 16 is  This proves the above first relation.

Squares from the Sums of Squares
For a natural number , let the sums > 1 k * 3 , : of powers be defined for We proceed with the version of the Theorem 9 for the sums .

Squares from the Products , and   
Let us introduce three binary operations , and  on the set of triples of integers by the rules and the operations , and are also the source of squares from components of the sixteen sequences Theorem 16.The following relations hold for the sequences A , :  This section contains four theorems which show that and 384 is the square of .
This proves the first relation.

34p
Proof.Since     , it follows that the difference Theorem 17.The following relations hold for the sequences A , , : p i. e., the square of 8 Theorem 18 The following relations hold for the sequences 2p .This concludes the proof of the first relation.
and 256 is

Squares from the Products and  
This section uses the binary operations and   defined by Proof.Since the sums   D are built from the Fibonacci numbers n F while the sequences , , and from the Lucas numbers n .Each triple in the sequences

Theorem 10 .
The following relations hold for the sums * 1 The following relations hold for the triples A , , : and .
This proves the first relation.Notice that, in our final result, the third added constant value is in all cases the complete square.Theorem 22.The following relations hold for the triples A , , : 