Discovering the Square Triangle Numbers: Evaluation (Math Problem Sample)

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DISCOVERING THE SQUARE - TRIANGULAR NUMBERS
Introduction
In the ancient world, the Greeks used to play with numbers one of their most favourite number games was that involving the use of polygons. Such number games were kind of interesting to an extent that mathematician decided to write a lot of literature about them. Organization of Polygonal numbers can be in such a way that they form polygonal matrices, made of positive integer elements. A triangle in this regard is a three-sided polygon while a square is a four-sided polygon.
Evaluation
Numbers like 1, 3, 6, 10, 15, 21, …. are triangular numbers (Lafer). They form an array of a triangular pattern. Also, note that the difference between any two succeeding triangular numbers increases by a given factor, so that prediction of the next integer can be possible. The patterns formed by triangular numbers are as shown in the figure below:
Figure 1: Triangular numbers CITATION Phi16 \l 2057 (Lafer)
On the other hand the numbers 1, 4, 9, 16, 25,… are commonly known as the perfect squares. The perfect squares have square roots which are integers. The patterns associated with perfect squares form squares as shown in figure 2 shown below:
Figure 2: Square numbers CITATION Ref12 \l 2057 (Refik and Karaatlı)
Other polygons like pentagons, hexagons, octagons, among others can also be given the same considerations to form number patterns. Triangular numbers form a sequence that can be expressed as 1, 3, 6, …., n(n+1)/2, ….. Some of these triangular numbers are also perfect squares. For example 1, 36, 1225,… are numbers that are both triangular and square since they satisfy all the conditions required in both cases.
To show that a given positive integer a is triangular, it is important to assign another integer m, so that the equation relates m and a by: a = m(m+1)/ 2. There exist two integers p and q were given by a= pq/2, where the absolute value of p-q =1. By coming up with the conjecture: Square-triangular numbers are provided by STn= pn2(pn+pn-1)2 where n is a set of natural numbers 1,2,3,….CITATION Phi16 \l 2057 (Lafer).
From this conjecture, it is evident that STn= pn2(pn+pn-1)2 forms a set of natural numbers. The conjecture turns out to be valid when proved. The theory turns out to be valid when proved. Square-triangular numbers are formed in such a way that any two positive integer’s q and r form a unique number expressed as q2r2 are related by the equation q2= 1+2y2. This equation is one of the Pell’s equations.
The method discussed above gives very dependable results for the triangle-square combination. One would be tempted to think of employing the same techniques of derivation to find the relationship between the square numbers and other polygons like a pentagon, octagons. Mathematicians are already working on the figures.
Conclusion
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