Generating Pythagorean Triples

Introduction

“Pythagorean triple, originated from the terminology referred to as Pythagorean Theorem, which states that each right-angled triangle has its sides that satisfy the formula x²+y²=z² and thereby, the 3 sides of a right-angled triangle are actually described by a Pythagorean triple” (Stillwell, 2003, p.5). Fundamentally, three numerals that are positive are comprised of in a Pythagorean triple. For instance, if e, f, and g are positive numerals, then e²+f²=g². In numerical stipulations, a Pythagorean triple is simply a set of 3 numerals x, y, z that “make-up the sides/lengths of a right-angled triangle” (Sierpinski, 2003, p. 7).

Idyllically, 3, 4, 5 are the smallest set of positive numerals in a Pythagorean triple. In this paper, someone has to complete reading Chapter ten from the textbook that is entitled, ‘Mathematics in Our World’, and which was written by Bluman G. Allan. In doing so, he/she will choose not less than five Pythagorean triples. Indeed, he/she ought to be concise on his/her reasoning. Thus, this paper aims at illustrating why the selected five groups of Pythagorean Triples are capable of working, specifically in the formula Pythagorean Theorem.

Generating Pythagorean Triples

There exists a formula that is uncomplicated, and which is able to engender Pythagorean triples. Ideally, the numerals 5, 4, and 3 are referred to as Pythagorean triples because 5²=4²+3². Similarly, the numerals 13, 12 and 5 are Pythagorean triples. This is in view of the fact that 13²=12²+5². Suppose j and k are two numerals that are positive, such that j is less than k, then k²-j², 2kj, and k²+m² form a Pythagorean triple (Bluman, 2005).

“Algebraically, it is easier to notice that the sum-of-squares of the first-two is similar to square of the last-one,” (Alperin, 2005, p. 809). In consequence, this technique is capable of engendering each triple. This simply means that a Pythagorean triple is a set of (x, y, z) that is capable of working in the equation x²+y²=z². By employing this formula, we are capable of finding any numeral of this kind, that is, “the square of x plus the square of y is equal to the square of z” (Alperin, 2005, p. 811).

By employing this formula, we can observe that Pythagorean triples that can be engendered, and which actually works encompass:

[3, 4, 5] i.e. 3²+4²=5²;

[8, 6, 10] i.e. 8²+6²=10²;

[15, 8, 17] i.e. 15²+8²=17²;

[27, 36, 45] i.e. 27²+36²=45²;

[33, 56, 65] i.e. 33²+56²=65²;

[45, 26, 53] i.e. 45²+26²=53²;

[13, 84, 85] i.e. 13²+84²=85²;

[20, 21, 29] i.e. 20²+21²=29²; and

[45, 108, 117] i.e. 45²+108²=117²

Conclusion

In a Pythagorean triple that is primitive, x and y are co-prime, that is, they do not share any numerals or prime factors. In this kinds of Pythagorean triples, either x or y can be an odd numeral, or an even numeral; basing on this, it can be seen that z will be odd (Eckert, 1992). Nonetheless, the notion of Pythagorean triples can be generalized by manifold ways, such as Pythagorean quadruple, Pythagorean n-tuple, Fermat’s Last-Theorem; and Heronian triangle-triples.

References

Alperin, C.R. (2005). Pythagoras Model Tree. American Mathematical Journal, 112 (8), 806-815.

Bluman, A. G. (2005). Mathematics In Our World. College of Allegheny: McGraw-Hill Publishers.

Eckert, E. (1992). Pythagorean Triples that are Primitive. College Mathematics Journal, 23 (4), 412-418.

Sierpinski, W. (2003). Pythagorean Triangles. London: Dover Publications.

Stillwell, J. (2003). Fundamentals of Number Theory. New York: Springer Publishers.