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We know already that the set of positive integers, which are satisfying the Pythagoras equation of three variables are called Pythagorean triples. The all unknowns in this Pythagorean equation have already been solved by mathematicians Euclid and

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International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Online International Journal Available at http://www.cibtech.org/jpms.htm 2012 Vol. 2 (4) October-December, pp.41-46/Kalaimaran Ara
Research Article
41
PROPER SOLUTION TO GENERATE SEQUENCES FOR SET OF PRIMITIVE PYTHAGOREAN TRIPLES Kalaimaran. Ara*
Construction & Civil Maintenance Dept., Central Food Technological Research Institute, Mysore-20, Karnataka, India *Author for correspondence
ABSTRACT
We know already that the set of positive integers, which are satisfying the Pythagoras equation of three variables are called Pythagorean triples. The all unknowns in this Pythagorean equation have already been solved by mathematicians Euclid and Diophantine. However, the solution defined by Euclid & Diophantine is also again having unknowns. The only possible to solve the Pythagorean equations is trial & error method and it will not be so practical and easy especially for time bound works, since the Pythagorean equations are having more than two unknown variables. The scope of work is to generate sequences of the primitive Pythagorean triples without missing any set of the triples, by single simple formula. After conducting various iterations, the author has developed a simple formula to generate plenty of sequences and plenty of terms in each sequence for set of primitive Pythagorean triples. The formula has been proved with an appropriate example. It is very useful for Students, Research scholars, Engineers and persons whose work is related to right-angled triangles, rectangular prisms and Number theory.
Key Words
: Right Triangle, Pythagoras Theorem, Number Theory and Primitive Pythagorean Triples.
INTRODUTION
The
Right angled triangles
(Weisstein Eric W, 2003, p.2575) are having significant role in the study of geometry. In any right triangle, one of the interior
vertex angles
(Weisstein Eric W, 2003, p.3152) is 90
. The longest of all three sides of the right triangle is called
hypotenuse
(Weisstein Eric W, 2003, p.1449). The right triangle has been used in trades for thousands of years. Ancient Egyptians found that they could always get a square corner using the 3-4-5 right triangle. Carpenters still use the 3-4-5 triangle to square corners. Later, a Greek mathematician named Pythagoras who lived about 2500 years ago developed the most famous formula to find the side lengths of any right triangles, possibly uses in all of mathematics. He proved that, for a right triangle, the sum of the squares of the two sides that join at a right angle equals the square of the third side, which is the side opposite the right angle is called the hypotenuse of the right triangle. The two shorter sides are usually called legs. The set of positive integers, which are satisfying the Pythagoras equation of three variables are called Pythagorean triples. The Pythagoras theorem has many uses in astronomical survey, trigonometry, application in Engineering and etc. Surveyors use it in their work. In any engineering work, the use of right triangle is very important. Using the
trigonometry
(Weisstein Eric W, 2003, p.3052), the airport cloud ceiling can be determined by the airport meteorologists to ensure the safety of the flights. The
primitive Pythagorean triples
(Weisstein Eric W, No date) are set of three numbers which are satisfying the Pythagorean equation as well as there is no common factors among them other than unity.
EXISTING FORMULAE AND METHODS
Pythagorean triples are a set of integers (x, y, r), which are satisfying the Pythagoras equation
. This equation is specifically applicable for right angled triangle (see fig.1). The unknowns in the Pythagorean equation have already been solved by mathematicians
Euclid and Diophantine
(Gellert. W Gottwald S, Hellwich. M Kästner H Küstner H, 1989, p.672). The solution defined by mathematicians Euclid & Diophantine for Pythagorean triples
as “All Pythagorean triples in which „x‟, „y‟, „r‟ are without common factor and „x‟ is odd are obtained by replacing the letters „a‟ and „b‟ in the triple
*
+
by whole numbers that have odd sum and no
common factor”.
However the solution defined by Euclid & Diophantine is also again having unknowns. However, the solution defined by Euclid & Diophantine is also again having unknowns. The existing methods are not much suitable to generate the complete set of primitive Pythagorean triples. The only possible to solve the Pythagorean equations was trial & error method. Moreover, the trial & error method to obtain these values are not so practical and easy especially for time bound works, since the Pythagorean equations are having more than two unknown variables. The method to generate Primitive Pythagorean triples was already defined by Fibonacci and Dickson.
International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Online International Journal Available at http://www.cibtech.org/jpms.htm 2012 Vol. 2 (4) October-December, pp.41-46/Kalaimaran Ara
Research Article
42
ANALYSIS AND FORMULATION FOR FORMULA
The primitive Pythagorean triples are set of three numbers which are satisfying the Pythagorean equation as well as there is no Greatest Common Divisor (GCD) (Weisstein Eric W, 2003
, p.1257
) among them other than unity. The values of integer numbers for x, y, r is a set of Pythagorean triples for Pythagorean equation x
2
+ y
2
= r
2
. The equation satisfies the right-angled triangle (see fig. 1).
Fig.1: A diagram of Right angled triangle ABC
Let (x, y, r) be a set of Pythagorean triples for Pythagorean equation x
2
+ y
2
= r
2
. After conducting several iterations by the author, the triples are sorted out depends upon its certain common properties such as
for each group. The set of primitive Pythagorean triples, which are available in online
“
http://www.tsm-resources.com/alists/trip.html
”
(Douglas Butler, iCT Training Centre of Oundle School, No date) are taken into account for this analysis. The mathematical relation between these value are studied carefully and these are sorted out into several groups depending upon the value of
and
. Let,
and
. Where, „m‟ and „n‟ are any natural integer numbers
. Finally these groups have been formulated to a single formula. The classified set of values of primitive Pythagorean triples (x, y, r) are tabulated as under: m
1 2 3 4 5 6 7 8 9 10 n
2 8 18 32 50 72 98 128 162 200
1
3 5 7 9 11 13 15 17 19 21 * 1
4 12 24 40 60 84 112 144 180 220 *
5 13 25 41 61 85 113 145 181 221 * 9
15 21 - 33 39 - 51 57 - 69 * 2
8 20 - 56 80 - 140 176 - 260 *
17 29 - 65 89 - 149 185 - 269 * 25
35 45 55 65 - 85 95 105 115 - * 3
12 28 48 72 - 132 168 208 252 - *
37 53 73 97 - 157 193 233 277 - * 49
63 77 91 105 119 133 - 161 175 189 * 4
16 36 60 88 120 156 - 240 288 340 *
65 85 109 137 169 205 - 289 337 389 *
International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Online International Journal Available at http://www.cibtech.org/jpms.htm 2012 Vol. 2 (4) October-December, pp.41-46/Kalaimaran Ara
Research Article
43
Table 1: Values of set of Pythagorean triples (x, y, r)
The above set of values refers that
and etc. Finally a formula which will be exactly appropriate to generate set of primitive Pythagorean triples at mth row and nth column for the Pythagorean equation
is formulated in matrix form as follows
(
)
√
√
√
,-
Where,
However, if there is a GCD other than unity for p and q of any particular set, it is not primitive triples.
Substituting values of 1≤
m
≤
various sequences can be developed. For each sequences substituting
1≤
n
≤
, the various elements of the sequence can be developed continuously.
81
99 117 - 153 171 - 207 225 - 261 * 5
20 44 - 104 140 - 224 272 - 380 *
101 125 - 185 221 - 305 353 - 461 * 121
143 165 187 209 231 253 275 297 319 341 * 6
24 52 84 120 160 204 252 304 360 420 *
145 173 205 241 281 325 373 425 481 541 * 169
195 221 247 273 299 325 351 377 403 429 * 7
28 60 96 136 180 228 280 336 396 460 *
197 229 265 305 349 397 449 505 565 629 * 225
255 285 - 345 - - 435 465 - - * 8
32 68 - 152 - - 308 368 - - *
257 293 - 377 - - 533 593 - - * 289
323 357 391 425 459 493 527 561 595 629 * 9
36 76 120 168 220 276 336 400 468 540 *
325 365 409 457 509 565 625 689 757 829 * 361
399 437 475 513 551 589 627 665 703 741 * 10
40 84 132 184 240 300 364 432 504 580 *
401 445 493 545 601 661 725 793 865 941 * 441
483 525 - 609 651 - - 735 777 - * 11
44 92 - 200 260 - - 392 464 - *
485 533 - 641 701 - - 833 905 - * 529
575 621 667 713 759 805 851 897 943 989 * 12
48 100 156 216 280 348 420 496 576 660 *
577 629 685 745 809 877 949 1025 1105 1189 *
√
* m
√
*
√
*
International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Online International Journal Available at http://www.cibtech.org/jpms.htm 2012 Vol. 2 (4) October-December, pp.41-46/Kalaimaran Ara
Research Article
44
m or n
1 1 2 2 9 8 3 25 18 4 49 32 5 81 50 6 121 72 7 169 98 8 225 128 9 289 162 10 361 200 11 441 242 12 529 288 13 625 338 14 729 392 15 841 450 16 961 512 17 1089 578 18 1225 648 19 1369 722 20 1521 800
…
…
…
…
…
…
…
…
…
Table 2: Values of for p and q for Pythagorean triples
These are the necessary procedure to generate any sequences of primitive Pythagorean triples. By using the formula (eqn.1), we can calculate plenty of set of primitive Pythagorean triples in each sequence. The GCD greater than unity are identified for specific values for ready reference as tabulated in table 3 for ready reference.
Note
: The symbol
refers that there is GCD other than unity for those two numbers and the symbol
refers that there is no GCD other than unity for those two numbers.
RESULT AND DISCUSSION EXAMPLE-1
The formulae for primitive Pythagorean triples (x
n
, y
n
, r
n)
for
and for
can be calculated by the formula:
2 8 1 8 3 2 5 0 7 2 9 8 1 2 8 1 6 2 2 0 0 2 4 2 2 8 8 3 3 8 3 9 2 4 5 0 5 1 2 5 7 8 6 4 8 7 2 2 8 0 0 8 8 2 9 6 8 1 0 5 8 1 1 5 2 1 2 5 0
1
9
25
49
81
121
169
225
International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Online International Journal Available at http://www.cibtech.org/jpms.htm 2012 Vol. 2 (4) October-December, pp.41-46/Kalaimaran Ara
Research Article
45
289
361
441
529
625
729
841
961
1089
1225
1369
1521
1681
1849
2025
2209
√
√
√
,-
Substituting
in above formula and
√
√
√
,-
Substituting n = 1 to 10, we get the values as in the form of matrix.
( )
Similarly, the formula for the other sequence can be generated.
CONCLUSION
In this article the existing formulae and method have been shown for the reference in order to understand the practical problem when solving the Pythagorean equation with primitive Pythagorean triples. A new formula to generate a sequence of primitive Pythagorean triples for which has been developed now have been defined and proved with two appropriate examples for detailed explanation. The eqn. 1 is the necessary formula for generating any sequences for set of Pythagorean triples. By the formula, we can generate plenty of sequences and each sequence is of infinitive terms of set of Pythagorean triples. A table of GCD for various values of p upto 2209 and q upto 1250 has also been provided. The formulae, which have been defined in this article will be one of the proper solutions to generate a sequence for set of primitive Pythagorean triples and also useful for those doing research or further study in the
Diophantine equations
(Eric Weisstein. W, 2003) and
Number theory
(Eric Weisstein. W, 2003) especially this formula will be as a clear road map to solve the problem of
Perfect Cuboid
(Eric Weisstein. W, 2012) which is having dimensions of all sides and diagonals with integers.
Acknowledgement
It gives immense pleasure to thank all those who extended their coordination, moral support, guidance and encouraged to reach the publication of the article. My sincere thanks to Prof. Ram Rajasekharan, Director, CSIR-

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