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International Refereed Journal of Engineering and Science (IRJES) ISSN (Online) 2319-183X, (Print) 2319-1821 Volume 2, Issue 7(july2013), PP.0-0 www.irjes.com

Pythagorean Triangle with Area/Perimater as a Special Polygonal Number M.A.Gopalan1, V.Geetha2 1

(Department of Mathematics, Shrimathi Indira Gandhi College, Trichirappalli.620 002) 2 (Department of Mathematics, Cauvery College For Women, Trichirappalli.620 018)

ABSTRACT:- Patterns of Pythagorean triangles in each of which the ratio Area/Perimeter is represented by some polygonal number. A few interesting relations among the sides are also given.

KEYWORDS:- Pythagorean triangles, Polygonal number MSC 2000 Mathematics Subject Classification:11D09 ,11D99

1. NOTATIONS Special Numbers Regular Polygonal number

Notations tm,n

Gnomonic number

Definitions

Carol number

carln

Star number Pronic

Sn Pn

22n  2n1  1

INTRODUCTION:

The method of obtaining non-zero integers x, y and z under certain conditions satisfying the relation

x 2  y 2  z 2 has been a matter of interest to various Mathematicians [1,2,3].In[4-15] special Pythagorean problems are studied. In this communication, we present yet another interesting Pythagorean problem. That is, we search for patterns of Pythagorean triangles where, in each of which the ratio Area/Perimeter is represented by a special polygonal numbers namely, Icosihexagonal, Icosiheptagonal , Icosioctagonal, Icosinonagonal, and Triacontagonal number by the symbols, X,E,Q,N and T respectively. Also, a few relations among the sides are presented.

II.METHOD OF ANALYSIS The most cited solution of the Pythagorean equation

x2  y 2  z 2

(1)

is represented by

x  2rs, y  r 2  s 2 , z  r 2  s 2 (r  s  0)

(2)

Case 1: Under our assumption

a = t 26,X p leads to the equation

s(r  s)  2 X (12 X  11)

This equation is equivalent to the following two systems I and II

rs

12 X  11 2X

s 2X 12 X  11

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Pythagorean Triangle With… In what follows , we obtain the values of the generators r, s and hence the corresponding sides of the Pythagorean triangle Pattern 1: On evaluation the values of the generators satisfying system I are

r  14 X  11, s  2 X

Employing (2), the sides of the Pythagorean triangle are,

x  56 X 2  44 X , y  192 X 2  308 X  121, z  200 X 2  308 X  121 Examples:

1 2 3 4 5 6

12 136 372 720 1180 1752

5 273 925 1961 3381 5185

13 305 997 2089 3581 5473

30 18564 172050 705960 1994790 4542060

30 714 2294 4770 81420 12410

Relations:

1.7 x  24 y  25z  121  0 2.( y  z  7 x  242)2  11858( z  y ) 3.3x  y  z can be represented by 44 times Decgonal number of rank X 4.8( y  z)  3(mod 28) 5.6( z 2  x 2 ) is a nasty number.

Pattern 2: On evaluation the values of the generators satisfying system II are

r  14 X  11, s  12 X  11

Employing (2) the corresponding Pythagorean triangle is

x  336 X 2  576 X  242, y  52 X 2  44 X , z  340 X 2  572 X  242 Examples:

1 2 3 4 5 6

6 442 1550 3330 5782 8906

8 120 336 656 1080 1608

10 458 1586 3394 5882 9050

24 26520 260400 1092240 3122280 7160424

24 1020 3472 7380 12744 19564

Relations:

1.84 x  132 y  85z  242  0 2.( x  13 y  z  484) 2  81796( z  x) 3. y  12t18, X  4t3, X 1 4.13x  y  13z  0(mod 44) 5.3( z  x) is a nasty number.

Case 2: Under our assumption

a = t 27 ,E p leads to the equation

s(r  s)  E (25E  23) www.irjes.com

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Pythagorean Triangle With… This equation is equivalent to the following two systems I and II

rs

25E  23 E

E 25E  23

As in the previous case, we obtain the values of the generators r, s and hence the corresponding sides of the Pythagorean triangle Pattern 3: On evaluation the values of the generators satisfying system I are

r  26E  23, s  E

Employing (2) the corresponding Pythagorean triangle is

x  52 E 2  46 E , y  675E 2  1196 E  529, z  677 E 2  1196 E  529 Examples:

1 2 3 4 5 6

6 116 330 648 1070 1596

8 837 3016 6545 11424 17653

10 845 3034 6577 11474 17725

24 48546 497640 2120580 6111840 14087094

24 1798 6380 13770 23968 36974

Relations:

1.52 x  675 y  677 z  1058  0 2.( y  26 x  z  1058)2  559682( z  y ) 3.x  y  z  4t56, E  4t3, E 1 (mod 6) 4.39 x  182 y  182 z  1196 t10, E

5.x  3 y  3 z  92t3, E 1

6.6( z 2  x 2 ) is a nasty number. Pattern 4: On evaluation the values of the generators satisfying system II are

r  26E  23, s  25E  23

Employing (2) the corresponding Pythagorean triangle is

x  1300 E 2  2346 E  1058, y  51E 2  46 E , z  1301E 2  2346E  1058 Examples:

1 2 3 4 5 6

12 1566 5720 12474 21828 33782

5 112 321 632 1045 1560

13 1570 5729 12490 21853 31818

30 87696 918060 3941784 11405130 26349960

30 3248 11770 25596 44726 69160

Relations:

1.1300 x  51y  1301z  1058  0

2.( x  51y  z  2116) 2  5503716( z  x) 3.x  102 y  z  2 E 2 (mod 23) 4. y 46 PE1  0(mod 5) www.irjes.com

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Pythagorean Triangle With…

5.x  216 S E  2( E  1)(mod 22 ) Case 3: Under our assumption

a = t 28,Q p leads to the equation

s(r  s)  2Q(13Q  12)

This equation is equivalent to the following two systems I and II

rs

13Q  12 2Q

2Q 13Q  12

As in the previous case, we obtain the values of the generators r, s and hence the corresponding sides of the Pythagorean triangle Pattern 5: On evaluation the values of the generators satisfying system I are

r  15Q  12, s  2Q

Employing (2) the corresponding Pythagorean triangle is

x  60Q 2  48Q, y  221Q 2  360Q  144, z  229Q 2  360Q  144 Examples:

1 2 3 4 5 6

12 144 396 768 1260 1872

5 308 1053 2240 3869 5940

13 340 1125 2368 4069 6228

30 22176 208494 860160 2437470 5559840

30 762 2574 5376 9198 14040

Relations:

1.60 x  221y  229 z  1152  0 2.(6( z  y )  45 x  1728) 2  583200( z  y ) 3.2 x  3 y  3 z  192t3,Q 1 4.15z  15 y  2 x  0(modQ) 5.50 y  48 z  t52,Q  0(mod 96)

6.2  x  96t3,Q 1  is a nasty number. Pattern 6: On evaluation the values of the generators satisfying system II are

r  15Q  12, s  13Q  12

Employing (2) the corresponding Pythagorean triangle is

x  390Q 2  672Q  288, y  56Q 2  48Q, z  394Q 2  672Q  288 Examples:

1 2 3 4

6 504 1782 3840

8 128 360 704

10 520 1818 3904

24 32256 320760 1351680

24 1152 3960 8448

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Pythagorean Triangle With… 5 6

6678 10296

1160 1728

6778 10440

3873240 8895744

14616 22464

Relations:

1.390 x  56 y  394 z  1152  0

2.( x  14 y  z  576) 2  112896( z  x) 3.14 x  y  14 z  0(mod 48)

4.z 2  x2  64Q2t16,Q 5.y can be represented by 8 times hexadecagonal number of rank Q Case 4: Under our assumption

a = t 29,N p leads to the equation

s(r  s)  N (27 N  25)

This equation is equivalent to the following two systems I and II

rs

27 N  25 N

N 27 N  25

As in the previous case, we obtain the values of the generators r, s and hence the corresponding sides of the Pythagorean triangle Pattern 7: On evaluation the values of the generators satisfying system I are

r  28 N  25, s  N

Employing (2) the corresponding Pythagorean triangle is

x  56 N 2  50 N , y  783N 2  1400 N  625, z  785 N 2  1400 N  625 Examples:

1 2 3 4 5 6

6 124 354 696 1150 1716

8 957 3472 7553 13200 20413

10 965 3490 7585 13250 20485

24 59334 61454 2628444 7590000 17514354

24 2046 7316 15834 27600 42614

Relations:

1.56 x  783 y  785z  1250  0 2.( y  28 x  z  1250)2  980000( z  y ) 3.x  28 y  28z  0(mod 50) 4.6 x  7 y  7 z  100t16, N

5.785 y  783z  1375GN  125  0 6.x(2n )  25  25carln (mod 31) Pattern 8: On evaluation the values of the generators satisfying system II are

r  28N  25, s  27 N  25

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Pythagorean Triangle With… Employing (2) the corresponding Pythagorean triangle is

x  1512 N 2  2750 N  1250, y  55 N 2  50 N , z  1513N 2  2750  1250 Examples:

1 2 3 4 5 6

12 1798 6608 14442 25300 39182

5 120 345 680 1125 1680

13 1802 6617 14458 25325 39218

30 107880 1139882 4910280 142324250 32912880

30 3720 13570 29580 51750 80080

Relations:

1.1512 x  55 y  1513z  1250  0

2.( x  110 y  z  2500)2  7562500( z  x ) 3.55x  y  55z  0(mod 5) 4.z  1250  135t24, N  t58, N (mod1373) 5.y can be represented by 5 times Icositetragonal number of rank N . Case 5: Under our assumption

a = t30,T p leads to the equation

s(r  s)  2T (14T  13)

This equation is equivalent to the following two systems I and II

rs

14T  13 2T

14T  13

As in the previous case, we obtain the values of the generators r, s and hence the corresponding sides of the Pythagorean triangle Pattern 9: On evaluation the values of the generators satisfying system I are

r  16T  13, s  2T

Employing (2) the corresponding Pythagorean triangle is

x  56T 2  52T , y  192T 2  364T  169, z  200T 2  364T  169 Examples:

1 2 3 4 5 6

12 152 420 816 1340 1992

5 345 1189 2537 4389 6745

13 377 1261 2665 4589 7033

30 26220 249690 1035096 2940630 6718020

30 874 2870 6018 10318 15770

Relations:

1.14 x  48 y  50 z  338  0 2.( y  7 x  z  338)2  16562( z  y ) 3.7 x  3 y  3 z  52t18,T www.irjes.com

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Pythagorean Triangle With…

4.2 x  y  z  208t3,T 1

5.x  32 y  31z  0(mod13) Pattern10: On evaluation the values of the generators satisfying system II are

r  16T  13, s  14T  13

Employing (2) the corresponding Pythagorean triangle is

x  448T 2  780T  338, y  60T 2  52T , z  452T 2  780T  338 Examples:

1 2 3 4 5 6

6 570 2030 4386 7638 11786

8 136 384 752 1240 1848

10 586 2066 4450 7738 11930

24 38760 389760 1649136 4735560 10890264

24 1292 4480 9588 16616 25564

Relations:

1.112 x  15 y  113z  338  0 2.4 y  5 x  5 z  52t22,T

3.( x  15 y  z  780) 2  151200( z  x) 4.226 x  15 y  225z  338  0 5.6( z 2  x 2 ) is a nasty number. CONCLUSION To conclude one may search for other patterns of Pythagorean triangles under consideration.

REFERENCES [1]. [2]. [3]. [4].

[9]

L.E. Dickson , History of the theory of Numbers, Chelsea Publishing Company, Newyork, Vol II,1952. Mordell L.J., Diophantine Equations Academic Press, Newyork, 1969. Carmichael, R.D., The Theory of Numbers and Diophantine analysis, Dover Publishings, Newyork, 1915. M.A.Gopalan and G.Janaki, Pythagorean Triangle with Nasty number as a leg, Journal of Applied Mathematical Analysis and Applications, Vol 4., No.1-2,Jan-Dec.,2008, 13-17. M.A.Gopalan and S.Devibala,On a Pythagorean Problem, Acta Ciencia Indica, VolXXXII M, No.4,2006, 1451-1452. M.A.Gopalan and S.Leelavathi, Pythagorean Triangle with 2 Area/Perimeter as a cubic iteger, Bulletin of Pure and Applied Sciences, Vol.26E,No.@,2007,197-200. M.A.Gopalan and G.Janaki, Pythagorean triangle with Perimeter as Pentagonal number, Antartica J Math, 5(2)2008,15-18. M.A.Gopalan and S.Vidhyalakshmi and N.Vanitha, Pythagorean equation and polygonal numbers, Impact J Sci Tech Vol2(3).2008,135-138. .M.A.Gopalan and S.Devibala, Pyhtagorean Triangle with Triangular Number as aleg. Impact J Sci Tech Vol2(4).2008,195-199.

[10].

M.A.Gopalan and Sriram , Pyhtagorean triangle with

[5]. [6]. [7]. [8].

[11]. [12]. [13] [14]. [15].

Area Perimeter

as a Difference of two squares, Impact J Sci Tech

Vol2(4).2008,159-167. M.A.Gopalan and G.Janaki, Pythagorean Triangle and Nasty number, Impact J Sci Tech Vol2(1).2008,37-42. M.A.Gopalan and G.Janaki, Pythagorean Triangle with Area/Perimeter as a special polygonal number,Bulletin of Pure and Applied Science, Vol.27E, No.2,2008,393-402. .M.A.Gopalan and A.Gnanam, A Special Pythagorean Problem, Acta Ciencia Indica Vol.XXXIII M, No.4, 2007,1435 Mita Darbari, Special Pythagorean Triangles with Perimeter as a sum of two squares and a cube, International Journal of Mathematical Sciences and Applications, Vol 1,No.3,September 2011,1225-1228. Mita Darbari, Special Pythagorean Triangles with Perimeter as a sum of two squares and a quartic International Journal of Mathematical Archive, Vol 2,No.7,July 2011,1094-1098.

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