 Advances in Pure Mathematics, 2011, 1, 286-294 doi:10.4236/apm.2011.15052 Published Online September 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Squares from D(–4) and D(20) Triples Zvonko Čerin Kopernikova 7, Zagreb, Croatia E-mail: cerin@math.hr Received May 5, 2011; revised May 20, 2011; accepted June 1, 2011 Abstract We study the eight infinite sequences of triples of natural numbers 2123 27,4 ,nnnAFF F, 21,nBF 25 274,nnFF, 21 2123,5 ,4nnnCF FF, 23 21,4 ,5nnDF F23nF and 21 23, 27,nnnL,4LL21,4LL25 27nnn, ,L21 21, 23nL,5nnLL,42321 23. The sequences and ,4LL,5nnnL,,ABCD are built from the Fibonacci numbers nF while the sequences , , and from the Lu-cas numbers n. Each triple in the sequences  L,,ABC and D has the property (i. e., adding 4D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.  20D Keywords: Fibonacci Numbers, Lucas Numbers, Square, Symmetric Sum, Alternating Sum, Product, Component 1. Introduction For integers , and c, let us write provided abbac2=ab c. For the triples =,,Xabc , =,,Ydef and =,,Xabc the notation YXX means that , and dbc aeca bfab c. When , let us write ,kk=,YkkXX for YXX. Hence, X is the Dk tri- ple (see  ) if and only if there is a triple X such that kXX. We now construct the infinite sequences A, , and of the Driples and , and  the Diples. They are B,4C23,FD4--trtof, AF2120 nn21, , , 27nF25 27,nnnF F21,4BF21,5FCFnn , and 23n4F23n23,4DF 21,5nnF F21,nL , , 27n234,nLL7n21 25,4nnLL2,L21,nL 2123 , 2321 23nnn, where the Fibonacci and Lucas sequences of natural numbers 4nnL5,LL,4LL,5nF and n are defined by the recurrence relations , , 2nL=10F=01F1=nnFFFL for and , , for . 2n2n0L=2 1L=1=n12nnThe numbers kLLF make the integer sequence from  while the numbers make . For an integer , let us use , 000045A000032AkLkkπk, and for , , kpkr2nkF2Lnk4Fnk and . 4LnkThe goal of this article is to explore the properties of the sequences A, , , , , , and . Each member of these sequences is an Euler BCD4D- or 20D-triple (see ) 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, , , , , , and . Most of our theorems have also versions for the associated sequences BCD 2π54,π=2πA,2 , 3,2π64,π=2πB, 2,22π,=2C 1, 23 2=2 ,,2πD and 54 22, ,2 , 64 3=2 ,,2, 221=10π,2 ,5π, 2=10π, 325π,2 that satisfy the relations 4AA20 4, , , and , , , . BB20 4CC4DD20 20 The overall principle in this paper is that if you can get complete squares by adding a fixed number to the prod-ucts 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 expres-sions 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 Z. ČERIN 287 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 de-tails. 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 BCD A, , , ,, , and become squares by adding some fixed integers. BCD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 prod-ucts 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 prod-ucts 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. 2. Squares from Products of Components The relations 4AA and imply that the com-ponents of 20 A and satisfy 423 1AAA and . Our first theorem shows that the product 2023 123 2 3116 AA is in a similar relation with respect to 9. Of course, the other products 31 3 1, 121 2 as well as 2323, etc. exhibit a similar property. The missing cases from the list coincide with the one of the previous cases. AAAABBTheorem 1. The following hold for the products of components: 9123 2 31023231223234111,,116 16 400AA BBCC pp ,p 641 031 3 183131431 3 161,,16AACC DDpp 5,p 901212 6121221, and 516 BB CCp p. Proof. Let 15=2 and 15 1==2. Since =jjjF and =jjjL, it follows that 232324=nnA, 273=A27nn and 223 23=4 nn, . 273=27nnAfter the substitutions 1= and =n, the sum of the product 23 2 3116 AA and 9 becomes 220 82085. However, this is precisely the square of . This shows the first relation. 10pThe version of the previous theorem for the sequences A, , , ,, , and is the following result. Notice that in this theorem there are no repetitions of cases. BCDTheorem 2. The products of the components of A,, satisfy: 1123 2 362323723233111,,444AABB CC   pp 1,r 1423 23531 3 1731319111,,41616  DDAA BB rp4,p 103131 33131 41212911 1,,100 804CCDDAA  pp 1,p 14121210121241 2 125111,,416100BB CC DD  pr 1.p Proof. Since 24 242=nn, 24 242=nnA, 22n223=2 n and 22 2232=Ann, the sum of 232 3 14AA and 1, after the substitutions 1= and =n, becomes 212 81285. However, the square of has the same value. This proves the first relation 6p123 2 36p14AA . The same kind of relations hold also for the products of components from four among the sequences A, , , ,, , and . BCD Theorem 3. The relations that hold for the products of components: 9023 231023 2362 3 2 361,16,02016 AB ACAD pp p , Copyright © 2011 SciRes. APM Z. ČERIN 288 111,p23238232382 323411 1,,256 400625BC BDCDpp 4,p 64 90313183 13 1103 13 16,,AB ADCDpp 9012 12612 12212 1 221,5,16 AB ACADpp 04.p Proof. Since 33 and 33, 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. =BA =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, , , ,, , and . BCDTheorem 4. The products of components of A,,satisfy: 11623 23723 2352 3 2 3611,,44AB ACAD pr 1,p 094r23234313183131 4114,, ,16 16CDAB AC pp 113131 53131 312129111,,16 1004BCCDAB   rr 1,p 491121271212 81212 9111,,16 44 ACAD BD pr6,r and 112 251100CD p. 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 3C2324 24nn , , and 21 21nn24 24nn21 215nn. It is now clear from the proof of Theorem 1 that the sum of 23 23 and 16 is precisely the square of 5. This requires the identities  ACr11 2π=p, 44 8π=p and . 25516=r28pp3. Squares from Symmetric Sums Let be the basic symmetric functions defined for 3123,,:=,,xabc by 12 3,, .xabc xbc caab xabc  Let *2, be defined by *31: *2=xbc caab and *1=xabc. Note that *1x is the determinant of the 1 3 matrix ,,abc (see  ). For the sums 2 and *2 of the components the following relations are true. Theorem 5. The following is true for the sums 2 of the components: 384 384868 622 22576 57650 83222 2128 809922 22336 32045222 298, 1114,74, 47, 4, 8, 8, 20.     ABCDABCDpprprp pppprp Proof. Since 924=2 35Br, 92=4 23r, the sum 2280B is 2916 2391r that we recog- 5nize as the square of . This proves the sixth relation 98p809228Bp. The sums *2B and *2 have constant values 4 and . On the other hand, , 200*322πA022*3, 64**2234Cr and . 644Dr**225Theorem 6. The following is true for the sums *2 of the components: 128 128**8 6**10822 2278,8 7AB ppp p,. 256 576**52* *60222 2813,4 3CD pp pp Proof. Since *1202=π23π7C and *1 402=23π12π, the sum is the **22256Csquare of 58132pp. This proves the third relation. Some similar relations make up the following two results. Theorem 7. The following is true for the sums *2 of the components: 80 16****6****32 22222224, 4,  AB BCpr and 16*** *5222 24.BD  rProof. Since , 2*32=2πA2*32=2, *2=4B and , it follows that *2=20Copyright © 2011 SciRes. APM Z. ČERIN 289 22** **33622 2280 =4π=4AB p . Theorem 8. The following is true for the sums 2 of the components: 14492222 4.AB r Proof. Since 924=65Ar, 92=4 6r, 924=2 35Br and 392=42r, it follows that 22 222229914416 16=3649144=455AB  rr9.r 4. Products of Sums as Differences of Squares The products of the sums 1 and *1 of the com- ponents of the four triples among A, , , ,, , and show the same kind of relations. This is also true for the associated triples BCD  A, , , ,, , and . 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 dif- ference of squares. BCDTheorem 9. The following relations hold for the sums 1: 64 400880p11 1111 111119 ,32 ,16 16AB AC  pr 225 129694 111111 1111114+11, +8,16 16AD BC pp rp 4 841 2511 041111 1111113, 3116 16BD CDpp p , 16 010 9811 1111 11113, 3,25 25AB AC  pp p 91977 61111 1111112, 9525 9AD BC ppp p 6, 49 9109 71111 1111111 ,2.9BD CD ppp Proof. The sums of the components 1A, 1, 1B and 1 are equal , 502π5025π, 90ππ and 90. Hence, the sum of 64 and 11 11116 AB is the square of . This proves the above first relation. 819pTheorem 10. The following relations hold for the sums *1: 01** **8** **611 1111 1114, ,64AB AC  pp 11**** 7**** 51111 111111,,16 64AD CD rr 16 16** **88** **8511 1111 11112,2 ,99AB AC  pr pp 25 256****75****10 31111 111112,3 ,9AD BC pp pr 529 9****10 4****41111 111125,11BD CD   pp p . Proof. The sums of the components *1A, *1, *1B and *1 are equal , 42π42, and 42π42. Hence, the product ** **11 11AB is the square of 84psince 44 8π=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**106** 1111 1111 1111,216 16AB AB  prpp 7, 25 64**40* *611 11111 1114, 264 16  5,AC ACrp rp81 49**92 **711 1111 11113, 35,216 16  AD ADppp p49 121**50* *12111 1111 1114,364 16  BC BD rpp p5,169 25**50** 8211 1111 11115, 216 64CD CD  rrp p . Proof. With the above information about the sums of the components 1A, 1, *1B and *1, the sum **111111616 AB is  850508 1050210 6π5π16 =5216=. pppprrp This proves the first relation. Theorem 12. The following relations hold for the sums 1 and *1: 16 16**88** 10911 1111 11112,3 ,25 9AB AB  prp p 16 0**65* *11 11111 1123, 925 AC AC   pp p 8, 25 9**75**911 1111 11112,2 725 9AD AD  ,pp pp Copyright © 2011 SciRes. APM Z. ČERIN 290 576 16**8* *811 11111 1111 ,34,9BC BC   p 7pp5, 1089 1** 77**1211 1111 1125, 3BD BD   prpr 25 121**95** 1011 1111 111,89CDCD  prp p 6. Proof. The sums of the components 1A, 1, *1B and *1 are equal , 45π45, 90ππ12 and 9012. Hence, the sum of the product 1*111125AB and 16 is 890 9 028189 0881ππ 1641=216=24 pppp ppr. This proves the above first relation. 5. Squares from the Sums of Squares For a natural number , let the sums >1k*3,:kk of powers be defined for =,,xabc by =kkkkxabc and *k=kkkxabcWe proceed with the version of the Theorem 9 for the sums . 2 of the squares of components. Theorem 13. The following relations are true for the sums 2: 224 22410 511422 221143, 7244AB pr pp, 416 41650 3222 21174, 7444CDrprp8,,8. 288 28810 511422 2243,72AB pr pp 672 67250 3222 274, 74CD rp rp Proof. Since 2A and 2 are 802272 185pp and , the difference of 80227218pp22A and 896 is equal2285212840, where  and  are 320767143451 5 and 438 253 567 5.10 But, one can easily check that this is the square of 586pr. This concludes the proof of the first relation. The next is the version of the Theorem 13 for the alternating sums *2 of the squares of components. Theorem 14. The following relations are true for the sums *2: 384 96**66**6 822 221119 ,25 ,4AB pr pr 24 416**51* *80222 2112,2 ,64 4CD rp rr 224 224**76**9 1022 227, 7AB rp rp,. 128 160**38* *6022228, 43CDrpp p Proof. Notice that the alternating sums of squares of components *2A and *2 are 502252 145rr and 502252 14rr . Hence, the sum of **22A and 384 is equal 2887798334875 51427798334875 5.50410 However, one can easily check that this is the square of 6119 6pr. This proves the first relation. Multiplied by five these products of the sums *2 of components show the same behavior. Theorem 15. For the sums *2, the following relations hold: 196 196**114**11822 225534, 31444AB ppp p, 44**82* *802222516,2,64 20CD pp pp 44**106** 12522 225511,545AB pr rp,3. 1444 1764**92**60222 25225,5 4CD rp rr Proof. With the above values of *2A and *2, the sum of **225A and 784 is equal 2887798334875 51427798334875 5.10082 But, this is the square of 11 468pp. This outlines the proof the first relation. 6. Squares from the Products , and  Let us introduce three binary operations , and  on the set of triples of integers by the rules 3,,,,=, ,,abcuvw aubvcw Copyright © 2011 SciRes. APM Z. ČERIN Copyright © 2011 SciRes. APM 291 ,,,,=,,,abc uvwavbwcu and the operations , and are also the source of squares from components of the sixteen sequences  A, . ,,,,=, ,.abc uvwawbucv Theorem 16. The following relations hold for the sequences A,: This section contains four theorems which show that  384 648911 1134 ,89,AB AC  pp 2p  74 1011 11156, 54AD BC rprp 1 47360,3716 1599  7411 1114 ,61,BD CD  rp 960 64  8511 1113,14 ,AB AC  pr  11 311 211 1115,23 ,4AD BC  rr pp 439 80559 561  13 4 411 116,37 ,BD CD  rrp  8711 11129 ,3,9AB AC 320 2563 pp r  1561 313610 38211 112,72 ,AD BC  rr pp  185 194 411 11112,9676BD CD pr p .of 11AB and 384 is the square of . This proves the first relation. 834pProof. Since 9011=47 525ABpp and , it follows that the difference 901=47 52pp Theorem 17. The following relations hold for the sequences A,,: 384 896**8 **811 11()()2,()()58AB AC  pp 4,r  ** **956 511 111,2100 AD BC prp p 9 24969, 321 1601** **9411311 1112, 274BDCD  prp p ,  128 48** **85 611 11174,234AB AC pp rp 5,  201 240** **86 8611 1113, 6,4AD BC rr rr  1121 561** **11 6411 1119,13 ,4BD CD  rr p Z. ČERIN 292  640 64** **10 5811 111107 ,,25AB AC pp p  ** **918 211 112, 4ADBC  pp pp 1177 1024,  ** **4411 1115, 4.16BD CD  pp 1105 16 Proof. Since the sums *1AB and *1 are 82225r* and , it follows that the sum 8222rof  *11AB and 384 is  22 288242238444 ,55 rr28p i. e., the square of 8Theorem 18 The following relations hold for the sequences 2p. This concludes the proof of the first relation. A,, : 256 1611 2116111112,316AB AC rpp r, 127 16811410 2111123,24AD BC pr pp 1,  721 16013 3411112, 15,4BD CD   pp p 1 40 3361111 111111,24AB AC pp ,p 92 20895 811117,3 2,4AD BC pp rp 11  91 10093 811111113,244BD CD,  pr pr 20 9881111115, ,416AB ACpp 12 120107 511111,444AD BCpp 1,p  5 281791111112, 5.36 4BD CD pr 2p Proof. Since the sums 1AB and 1 are 4301π59 π9π2 and 43159 920, it fol- lows that the sum of 11AB and 256 is 83030159π9π59 92564p, i. e., 288787407352139 5638787407352139 54070440 which is the square of 11 22rp. This is the outline of the proof of the first relation. Theorem 19 The following relations hold for the sequences A,,: 256 11457 2****111112,2,16AB AC pr pp 10 511 3***11114,2 ,4AD BC pr pp 129 236*1  12 572***11117, 2,4BD CD  pp rr 945 4*1 Copyright © 2011 SciRes. APM Z. ČERIN 293 44 20888****111113,3 2,4AB ACpp 2r 10 270***11112,9 ,4AD BC pp pp 72 144*1  11 29 2****111111,544BD CD   ppp p 59 100, 54****111111,,416AB AC pp 828 93 8***1111,,4AD BC pr p 224 204*1  84***111112, .425BD CD   pp 69 1*1Proof. Since the sums *1AB and 1 are 4301π53π7π2 and 07431532**11, it follows that the sum of AB and 256 is 307256830153π7π534p, i.e., 288629487281515 5638629487281515 54070440 which is the square of . This is the outline of the 14 52pproof of the first relation. 7. Squares from the Products and  This section uses the binary operations and  defined by  ,,,, =,,,,,,, =,,.abcde fbfcecdafaebdabcde fbfcecdafaebd Note that restricted on the standard Euclidean 3 -space the product is the familiar vector cross-product. 3Theorem 20. The following relations hold for the triples A,, : r9068111111,,16 64AA BB rp 22111 111,.64 16CC DD  rr 49 Proof. Since the sums 1AA and 1 are 244π and 4220π, it follows that the sum of 11116AA  and 9 is , i.e., the 485pp 96rsquare of . This concludes the proof of the first relation. Theorem 21. The following relations hold for the triples A,,: 71 186 9** **11 111145,16 1024AA BB  pp p ,  476 7992*82** *111 111,( )25416  CC DDrrp p. Proof. Since the sums 1AA and 1 are 8644575rr and , it follows that the difference of the product 8644 5rr7**11116 AA and 71 is Copyright © 2011 SciRes. APM Z. ČERIN 294 28614 54045rr  which simplifies to 2883854117236 51013854117236 551005 i.e., to the square of 8645pp. 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,, :  22848811 11119 ,139 ,4AB AC  pp 02p  30 3682 10211 11179,4 34AD BC  rpp p 22,   29 510 6411 111118,31 ,44BD CD  rp p 22  848811 11121,320 ,4AB AC  pp 2285r  6793 8111 1113,11 6,25 AD BC  rp rp 222  27 1910 6411 11126,63 .4BDCD  rp p 22 Proof. Since the sums 1A and 1B are 11 423 2pr and , it follows that the sum 14 8212ppof the product 1114AB and 64 is which simplifies to the square of . This proves the first relation. 1141483 21264prp p819p8. References  E. Brown, “Sets in Which xyk Is Always a Square,” Mathematics of Computation, Vol. 45, No. 172, 1985, pp. 613-620.  N. Sloane, On-Line Encyclopedia of Integer Sequences. http://www.research.att.com/ njas/sequences/.  L. Euler, “Commentationes Arithmeticae I,” Opera Om-nia, Series I, volume II, B.G. Teubner, Basel, 1915.  Z. Čerin, “On Pencils of Euler Triples I,” (in press).  Z. Čerin, “On Pencils of Euler Triples II,” (in press).  M. Radić, “A Definition of Determinant of Rectangular Matrix,” Glasnik Matematic, Vol. 1, No. 21, 1966, pp. 17-22. Copyright © 2011 SciRes. APM