1.

The mathematical form of the evenly even and the oddly even numbers: , where 0, 1,2, 3 ? r .

Table 1:

In a right-triangle, the square of hypotenuse is equal to the sum of the squares of the other sides called the legs. This is known as the theorem of Pythagoras [1]. These three measurements are known as Pythagorean Triplets or Pythagorean Triples.

For the positive real numbers, , , and a b c , which are the measures of three sides of a right triangle the Pythagorean Theorem is 2 2 2 a b c ??, c is the measure of the hypotenuse.

We will discuss the known formulas in terms of evenly even, oddly even and odd numbers to determine the pattern.

Here we discuss some well-known results on Pythagorean triplets ( , , ) abc , Proposition 2. Suppose a is an evenly even number, then The following results are Pythagorean Triples ( , , ) abc , a is an odd number, Proposition 1. (3,4,5) (

We propose a method to find the Pythagorean primes using the magic rule 8 4 12 ?? .

Magic rule 8 -4 -12: Choose a Pythagorean prime, then add 8, 4, or 12 in order to collect the next Pythagorean prime. In this process one needs to observe the output. If the output is not a prime, filter it or mark it and continue with the process. Given below is an elastration step by step. Continuing in this process one may easily find infinitely many Pythagorean primes.

We further check the following:

It is interesting to note that each Pythagorean prime number is the sum of one even squares and one odd squares congruent to 1 mod 4 and this representation is unique.

Following list shows the prime positions of Pythagorean primes. The Pythagorean primes are in row 1, their prime positions are in row 2.

, then to evaluate the definite integral

It is expected that the readers know how to simplify the integrand.

It is not too difficult to check that

, which is a Pythagorean.

One has to verify that 21 ? ab or 41 ? ab .

London Journal of Research in Science: Natural and Formal

Definition. Hinges are positions when we split an ordered data set into pieces. John Tukey's upper hinge and lower hinge are the measures of positions, known as third and first quartiles.

Let 0 p ? be a positive integer. Two integers N and r are congruent modulo p, if there is an integer 0, 0 1 k r p ? ? ? ? such that N r kp ?? , and commonly known by the notation mod N r p ? Notice that the condition " N r kp ?? " for some integer k" is equivalent to the condition "p divides Nr".

Suppose we have N discrete ordered data points and to find p segments keeping m data points in each segment. The number of hinges must be 1 p ? .

To determine how many hinges are integer ranked and how many are non-integer ranked.

Nr m pp ??, the number of data points in each segment is Nr m p ? ? and there are r integer ranked hinges.

For simplicity we discuss a special case for four equal divisions commonly known as quartiles [4], and the same idea is extended for deciles [2] and further on even order of divisions or segments.

Suppose N is an even number, the middle most hinge will be non-integer ranked.

Further if N is doubly even, the first and third hinge are non-integer ranked as well. The number of data points N is divisible by 4. This result is confirmed by the remainder rule * , 0; , 4 N m r m r Z ? ? ? , there is no integer ranked hinges.

If N is singly even, then the remainder is 2 when N is divided by 4. The middle most hinge will be non-integer ranked and the other two must be integer ranked. The first hinge therefore is 1) m ? th data point and the third one is () Nr ? th data point, [4], [5].

Corollary 1: If the divisor p is an even number, then there exist midhinge (median) and data set shows symmetry about midhinge.

H is considered as the median of the ordered data set [4]. If p is odd, midhinge does not exist for the ordered data set and there is no symmetry.

Corollary 2: If the number of data points N is divisible p and ,0 N mp r ?? , then there is no integer ranked hinge. The positions of the hinges would be between each consecutive groups of m observations.

Note that if N is an odd number and 4 p ? , then the remainder is either 1 or 3. On the other hand if N is an even number then remainder is either 0 or 2. Remainder 1 r ? confirms the middle most hinge (median) as integer ranked and other two non-integer ranked keeping m data points in each segment. Thus we have the following table using average:

The remainder rule we propose works for hinges when the divisor p is an even number. But the remainder rule still works when p is an odd number. In this case, the number of hinges is even, which shows an interesting behavior. Finding integer ranked hinges we keep as an open question.

For example, we have 22 ordered data points and to find 8 hinges for nine segments.

We have 22, 9 Np ?? , therefore 22 4 mod 9 9 ? , where 4, 2 rm ?? . It is not difficult to verify that there are 4 integer-ranked hinges and remaining 4 hinges are non-integer ranked. The number of data points in each segment is 2 m ? .

Following are the possible selections. Suppose there are N elements in an ordered data set, we are interested to find 1 f ? fractiles. In this model f is an even number, [2], [4] .

The ? -th fractile is calculated as follows

, where i and df ? are positive integers.

The following model produces fractiles by the following rounding notion: In this example all the fractiles are non-integer ranked, with 8 ordered data points in each segment.

Position 8 th -9 th 16 th -17 th 24 th -25 th 32 nd -33 rd 40 th -41 st 48 th -49 th 56 th -57 th In this paper, we proposed several methodologies to solve complex mathematical problems, portraying pattern recognition and parity which could make complex math problems easier. We proposed strategies to determine powers of imaginary roots, calculating Pythagorean t riplets and Pythagorean primes using our "Magic Rule 8-4-12", and finding arc lengths and fractiles. Applying these methods could enhance students' background knowledge and skills to face the challenges in STEM education. These methods would be further studied to determine if students are able to implement these tactics to solve mathematical problems more efficiently.