Pythagorean triples refer to the mathematical expression that involves the desired integer definite solutions to the Pythagorean Theorem. It normally expressed utilizing the variables in the form of a2 + b2 = c2 (Benjamin & Gerhard, 2007). This mathematical expression called the Pythagorean triples because it normally involves the three variables that are correlated and interrelated.
Pythagorean Triple #1
The Pythagorean triples formula is normally derived utilizing the right angle triangle having the variable c side standing for the hypotenuse on the reverse side of the right angle. The side a is considered as the shorter section relative to the dual adjacent side of the chosen right angle triangle. The first and the foremost rules that are normally considered are becoming aware of the means of determining the subset of the Pythagorean triplets following the stipulated rules. The rule number one is through considering the side a of a Pythagorean triplet, the side b of a Pythagorean triplet is taken and expressed as (a2 – 1) / 2 and the third side c is b+1
Within this section, we will take a and c to be always odd variables and the b as ever even variables. The associations hold that the divergence amidst the consecutive square numerical figures taken as the consecutive of the odd numerical figures. Each odd numerical figure that is presented as square in order to make the derivation of the proper Pythagorean triplet.Therefore, the square of 7 is 49 and the represented the difference amidst the 576, which is the square of the 24 and the 625,the square of 25 hence giving the result as pertaining to the Pythagorean triplet as 7,24,25. Conversely, the square of 23 is depicted as 529 that is in turn represented as the difference amidst as 69696, which is the square of 264.The square of 265 is depicted as 70225 thus giving the result in term of the Pythagorean triplet as 23,264,265. Utilizing the break Fermat`s Last Theorem, the simplest triplet within the table, the values are not representing the right angle triangle. The variable a>b normally represents the definite solution of the Pythagorean theorem express using the formula m,0,m, where the variable m represent ant number.
Considering on the Euclid`s verification the infinite numerical of Pythagorean triplets ideally because the countless figures are odd figures. An elegant confirmation of the theorem is thus depicted on the illustration within the completely square represented as (a+b) 2, the area of every triangle is ½ ab and the area of the internal square is c2. The regional area of the whole square therefore also represented as 4(1/2 ab) + c2 hence
(a+b)2 = 4*(1/2 ab) + c2. When this expanded we get a2 + 2ab + b2 = 2ab + c2 and further simplification the result is a2 + b2 = c2 that represent the required Pythagorean theorem.
This can be proved via looking at the successive difference squares odd numerical figures in line with the required Pythagorean Theorem utilizing the summation of the each dual successive triangular numerical figure, which is depicted by the whole integers. Applying the summation represented as 0+1=1, 0 + 1 + 2 = 3, 0 + 1 + 2 + 3 = 6, 0+1+2+3+4= 10, 0 + 1 + 2 + 3+4+5 = 15 in that order. The subsequent figures are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, and 120. Taking the summation of the last two values that is 105+120 = 225 that represents a square of numerical figure 15.Each square is then divided into two sectional triangles thus every square numerical as mapped on the figure, which is sub, divided into triangles of dimension 10 and 25.
Pythagorean Triple #2
Taking the Pythagorean, triplets of side a being even number in deriving formula, an infinite series of the Pythagorean solutions through following the rules stipulated. Each side a to have an even number of the Pythagorean triplets, the b side in that regard is represented in the form of a/2)2 – 1 and then also considering the third side c to be depicted as b+2. The Pythagorean table stipulates that for any triplet, which is not evenly divisible by the number four normally represent even since such figures are readily multiples of two as depicted in the preceding evaluation above. The general formulas for all Pythagorean triplets is represented in the form of
a = p * q
b= (p2 – q2)/2
c = (p2 + q2)/2
Where m and n are the odd integers and p>q having no ordinary divisors. This formula only works for certain problems
Alternatively the version of the equations that can be utilized is
a = 2pq
b = p2 – q2
c = p2 + q2
The above apply when all the values of p and q are both not odd numbers. The table above for computation is derived via application of [n, 1] where n is an odd number bigger than one. The entire solution is derived via application of the general law hence the series 3, 4, 5 is represented as either [3, 1] or [2, 1] despite the stipulated sequence of [2, 1] which is resented as 4, 3, 5. The triplet of numbers 16, 63, 65 in that sequence utilizing the second law is consequent as [8, 1] and subsequently as the [9, 7] using the first law (Reinhard & David,2000). Thus, the non-primitive triplets that exist within the second tabulation via breaking the law. Considering the [4, 2] since it compose of even numbers which is contrary to the second rule but does give the result as expected triplet 8,6,10 hence reduced to 4,3,5. This is consequential from the reduction of the numbers within the bracket by a common factor of two thereby the final triplet series does not depend on the table by producing 20,21,29 from the dual numbers [7,3] using the first rule and [5,2] via application of the second rule.
By checking the result of the Pythagorean Theorem, we follow the following steps.
Pythagorean Triple #3
We can generate Pythagorean theorem via using three dimension via application of the third power such as in the form of x2 + y2 + z2 = r2 thus named the Pythagorean quadruplet(James,2005). This investigated using the existing triplet such as within the triplet 52 + 122 = 132. 52 can be examined and presented in the form of 32 + 42. Hence, the final Pythagorean quadruplet is 32 + 42 + 122 = 132.
Pythagorean Triple #4
There also exist a four dimensional Pythagorean Theorem via application of the separation formula. This formula is presented as t2 – (x2 + y2 + z2)/c2 = s2.
Where t = time, s= distance and c= velocity of light
Within this formula, the stipulated values are not computation since the velocity of light, time and distance is given. It can easily computed if when the integral values in places of the coordinates x2 + y2 + z2 is equal to the value of r2 and with the expression t2 – r2 = s2, the resulted version of the Pythagorean Theorem is called Pythagorean quintuplet.
Example of the application of the formula is
32 + 42 + 122 + 842 = 852
Where c = 1 and r = radius
t2 – (x2 + y2 + z2) = s2 as 852 – (32 + 42 + 122) = 842.
Pythagorean Triple #5
The association of the three above theorems is represented using Fermat by giving the following summary. Every prime numerical figure taking the form of 4n + 1 is depicted as the summation of the two squares (Edward, 1997). The prime number will always be the hypotenuse of Pythagorean triplets that is the addition of the dual squares. The third summary is that the hypotenuse of the dual Pythagorean triplets is represented by its squares and the cubes of the sides normally represent the hypotenuse of three Pythagorean Triplets. Hence,
I. 5 = 22 + 12,
II. 52 = 32 + 42,
III. 54 = 252 = 152 + 202 = 72 + 242, and
IV. 56 = 1252 = 752 + 1002 = 352 + 1202 = 442 + 1172.
Generally, Pythogorean theorem is derived from the Euclid and the feature of Pythagorean triplets has been employed in the numerous application s. It is normally applied in the geometrical analysis of the number theory. The formula relates the lengths of any existing triangles and the cosine rule and the vectors.
References
Laubenbacher, R., & Pengelley, D. (2000). Mathematical expeditions: Chronicles by the explorers. New York: Springer.
Fine, B., & Rosenberger, G. (2007). Number theory: An introduction via the distribution of primes. Boston: Birkhäuser.
Barbeau, E. (1997). Power play. Cambridge: Cambridge University Press
Tattersall, J. (2005). Elementary number theory in nine chapters. Cambridge: Cambridge Univ.Press.
