## 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 x*^{2}*+y*^{2}*=z*^{2}* 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 e^{2}+f^{2}=g^{2}. 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).

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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 5^{2}=4^{2}+3^{2}. Similarly, the numerals 13, 12 and 5 are Pythagorean triples. This is in view of the fact that 13^{2}=12^{2}+5^{2}. Suppose j and k are two numerals that are positive, such that j is less than k, then k^{2}-j^{2}, 2kj, and k^{2}+m^{2} 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 x^{2}+y^{2}=z^{2}. 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. 3^{2}+4^{2}=5^{2};

[8, 6, 10] i.e. 8^{2}+6^{2}=10^{2};

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[15, 8, 17] i.e. 15^{2}+8^{2}=17^{2};

[27, 36, 45] i.e. 27^{2}+36^{2}=45^{2};

[33, 56, 65] i.e. 33^{2}+56^{2}=65^{2};

[45, 26, 53] i.e. 45^{2}+26^{2}=53^{2};

[13, 84, 85] i.e. 13^{2}+84^{2}=85^{2};

[20, 21, 29] i.e. 20^{2}+21^{2}=29^{2}; and

[45, 108, 117] i.e. 45^{2}+108^{2}=117^{2}

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## 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.