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 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 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 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 Proof: With respect to either r is even
or odd, according to given expressions, it is straightforward to verify that 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.
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
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 For 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 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 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. REFERENCES Juan B. Gil and Aaron Worley, Generalized Metallic Means, Fibonacci Quarterly, Volume 57 (2019), Issue. 1, 45-50. Retrieved from https://arxiv.org/abs/1901.02619 K. Hare, H. Prodinger, and J. Shallit, Three series for the generalized golden mean, Fibonacci Quart. 52(2014), no. 4, 307-313. Krcadinac V., A new generalization of the golden ratio. Fibonacci Quarterly, 2006;44(4):335-340. Retrieved from https://arxiv.org/abs/1401.6200v1 P.N. Vijayakumar, R. Sivaraman, Probing Pythagorean Triples, Proceedings of 6th International Conference on Advanced Research in Arts, Science, Engineering and Technology (ICARASET - 2021), DK International Research Foundation, ISBN: 978-93-90956-53-1, pp. 39 - 41, 2021. Retrieved from https://www.researchgate.net/profile/Dr-Ishan-Pandya/publication/353121277_DISEASE-X_CROWN_VIRUSES_TRANSFORMED_LIVESINTO_CORONA_FACTORIES_AND_DANCE_OF_THE_PANDEMIC_COVID-19/links/60e820950fbf460db8f30c59/DISEASE-X-CROWN-VIRUSES-TRANSFORMED-LIVESINTO-CORONA-FACTORIES-AND-DANCE-OF-THE-PANDEMIC-COVID-19.pdf R. Sivaraman, Exploring Metallic Ratios, Mathematics and Statistics, Horizon Research Publications, Volume 8, Issue 4, (2020), pp. 388 - 391. Retrieved from https://doi.org/10.13189/ms.2020.080403 R. Sivaraman, Generalized Lucas, Fibonacci Sequences and Matrices, Purakala, Volume 31, Issue 18, April 2020, pp. 509 - 515. R. Sivaraman, Pythagorean Triples and Generalized Recursive Sequences, Mathematical Sciences International Research Journal, Volume 10, Issue 2, July 2021, pp. 1 - 5. R. Sivaraman, Relation between Terms of Sequences and Integral Powers of Metallic Ratios, Turkish Journal of Physiotherapy and Rehabilitation, Volume 32, Issue 2, 2021,1308 - 1311.
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