I now present ways to generate Pythagorean triples through binomial coefficients which are entries of Pascal’s triangle through the following theorems.

 

3.    Theorem 1

Let . Then for , the Pythagorean triple (a, b, c) is given by

Proof: Using the definition of the binomial coefficient from (2.2), we have

Thus, (a, b, c) forms Pythagorean triple and this completes the proof.

 

4.    Theorem 2

For , let , Then the Pythagorean triple (a, b, c) is given by

Proof: First, we notice that

Thus, (a, b, c) forms Pythagorean triple and this completes the proof.

In the following theorem, I will present a more general result to generate Pythagorean triples from particular entries of Pascal’s triangle.

 

5.    Theorem 3

Let , and let  be two natural numbers. Then the Pythagorean triple (a, b, c) is given by

 

 

 

Proof: From the given values of a, b we see that

Since a, b and c are integers, from (2.1), it follows that (a, b, c) is a Pythagorean triple. This completes the proof.

 

6.    Theorem 4

For any two integers , the primitive Pythagorean triples are given by

 if r is odd and

 if r is even.

Proof: With respect to either r is even or odd, according to given expressions, it is straightforward to verify that .  Moreover, from (6.2), if r is even, then it is clear that a, b, c are integers. If r is odd, then   is clearly an odd integer. Now the numerator of b and c namely and becomes even since if r is odd,  is odd. Hence b and c according to the expressions given by (6.1) must be integers. Moreover, the greatest common divisor of a, b, c is 1. Hence in either case, (a, b, c) forms a primitive Pythagorean triple. This completes the proof.

 

7.    Illustrations

In this section, I present some illustrations to generate Pythagorean triples as discussed in the three theorems and corollary in sections 3 to 6.

 

Figure 1 Pascal’s Triangle with first two entries highlighted beginning with first row

                                                                                     

As in theorem 1, if we consider the first two entries of Pascal’s triangle beginning from row 1, as shown in Figure 1, we can generate the following Pythagorean triples using (3.1)

For n = 1, we have  

For n = 2, we have  

For n = 3, we have  

For n = 4, we have  

For n = 5, we have  

For n = 6, we have  

For n = 7, we have  

 

Figure 2 Pascal’s Triangle with second and third entries highlighted beginning with second row

                                                                                                     

As in theorem 2, if we consider second and third entries of Pascal’s triangle beginning from row 2, as shown in Figure 2, we can generate the following Pythagorean triples using (4.1)

For n = 2, we have  

For n = 3, we have  

For n = 4, we have  

For n = 5, we have  

For n = 6, we have  

For n = 7, we have  

For n = 8, we have  

 

 

 

As illustration of equation (5.1) of theorem 3, if we choose the entry 35 located in seventh row (see Figure 2), we have . Here r = 3 or 4.

For  using (5.1), we can generate the following Pythagorean triples of the form .

Since r = 3 is odd, by (6.1), the primitive Pythagorean triples are given by

For k = 2, we have

For k = 3, we have  

For k = 4, we have

Similarly, for  using (5.1), we can generate the following Pythagorean triples of the form .

Since r = 4 is even, by (6.2), the primitive Pythagorean triples are given by

For k = 2, we have

For k = 3, we have

For k = 4, we have

Similarly, by choosing any particular entry from Pascal triangle and finding corresponding value of r, we can determine several Pythagorean triples.

 

8.    Conclusion

The main purpose of this paper is to connect the concepts of generating Pythagorean triples with the entries of Pascal’s triangle.

In theorem 1, by choosing the first two entries of the Pascal’s triangle and beginning from row 1 as shown in Figure 1, I had generated Pythagorean triples given by (3.1). Similarly, in theorem 2, beginning with second row, choosing second and third entries as shown in Figure 2, I had generated Pythagorean triples using the formulas given by (4.1).

In section 5, in theorem 3, by choosing any entry of Pascal’s triangle located at third place or higher place in a particular row, I had provided formulas for generating Pythagorean triples through equation (5.1). It is to be noted that the triples generated through (5.1) are not primitive. To generate primitive Pythagorean triple, I had provided theorem 4, in section 6, in which depending on the parity of r, two formulas are provided through equations (6.1) and (6.2). In particular, if r is odd, then the primitive Pythagorean triple is given by (6.1) and if r is even, then they are given by (6.2). Thus by choosing different values of  we can generate as many primitive Pythagorean triples as possible. In section 7, several illustrations were provided to explain the formulas obtained in theorems 1 to 4 of this paper. These calculations provided various Pythagorean triples as expected through the theorems established.

Thus, by proving four new theorems in this paper, I had exhibited the connection between the entries of Pascal’s triangle and generation of Pythagorean triples, the two fascinating and everlasting concepts in mathematics.

 

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