Because of the growing role of Binomial Theorem in various fields in Mathematics, such as in calculus and number theory, and even in societal advancements, such in technology and business, mathematicians continue to explore new developments in this interesting theorem. In this case, this study explored some of the unveiled concepts of Binomial Theorem for further studies, specifically on observing the relationships of the indices of the binomials to obtaining odd-valued binomial coefficients. Using the proof by exhaustion, among various mathematical proofs, a formula for n of the binomial in which the cases are separated and focus on the condition is supported. After a series of exhaustion of data, cases were determined and identified. Among the 64 values tested consecutively from 0 to 63, it was found that only the values of 0, 1, 3, 7, 15, 31, and 63 for n resulted to all odd binomial coefficients. Through trial-and-error process, the general term of the indices was expressed in the form of n = 2r – 1, Where r∈Z-*.
Cite this paper
Campos, E. M. and Decano, R. S. (2022). On the Study of Binomial Theorem: Formulation of Conjecture for Odd Binomial Coefficients for Binomials with Indices of n = 2r – 1, Where r∈Z-*. Open Access Library Journal, 9, e8670. doi: http://dx.doi.org/10.4236/oalib.1108670.
Echi, O. (2006) Binomial Coefficients and Nasir al-Din al-Tusi. Scientific Research and Essays, 1, 28-32.
https://academicjournals.org/journal/SRE/article-full-text-pdf/7A2190112547.
Khmelnitskaya, A., van der Laan, G. and Talman, D. (2016) Generalization of Binomial Coefficients to Numbers on the Nodes of Graphs. Discussion Paper Vol. 2016-007, CentER (Center for Economic Research), Tilburg.
https://doi.org/10.2139/ssrn.2732524
Flusser, P. and Francia, G.A. (2000) Derivation and Visualization of the Binomial Theorem. International Journal of Computers for Mathematical Learning, 5, 3-24.
https://doi.org/10.1023/A:1009873212702
Salwinski, D. (2018) The Continuous Binomial Coefficient: An Elementary Approach. The American Mathematical Monthly, 125, 231-244.
https://doi.org/10.1080/00029890.2017.1409570
https://www.tandfonline.com/doi/abs/10.1080/00029890.2017.1409570.
Zhu, M.H. and Zheng, J. (2019) Research on Transformation Characteristics of Binomial Coefficient Formula and Its Extended Model. Journal of Applied Mathematics and Physics, 7, 2927-2932. https://doi.org/10.4236/jamp.2019.711202
Che, Y. (2017) A Relation between Binomial Coefficients and Fibonacci Numbers to the Higher Power. 2016 2nd International Conference on Materials Engineering and Information Technology Applications (MEITA 2016), Qingdao, 24-25 December 2016, 281-284. https://doi.org/10.2991/meita-16.2017.58
Lundow, P.H. and Rosengren, A. (2010) On the p, q-Binomial Distribution and the Ising Model. Philosophical Magazine, 90, 3313-3353.
https://doi.org/10.1080/14786435.2010.484406
https://www.tandfonline.com/doi/abs/10.1080/14786435.2010.484406.
Su, X.T. and Wang, Y. (2012) Proof of a Conjecture of Lundow and Rosengren on the Bimodality of p, q-Binomial Coefficients. Journal of Mathematical Analysis and Applications, 391, 653-656. https://doi.org/10.1016/j.jmaa.2012.02.049
Usman, T., Saif, M. and Choi, J. (2020) Certain Identities Associated with (p, q)-Binomial Coefficients and (p, q)-Stirling Polynomials of the Second Kind. Symmetry, 12, Article No. 1436. https://doi.org/10.3390/sym12091436
Gavrikov, V.L. (2018) Some Properties of Binomial Coefficients and Their Application to Growth Modelling. Arab Journal of Basic and Applied Sciences, 25, 38-43.
https://doi.org/10.1080/25765299.2018.1449346
https://www.tandfonline.com/doi/full/10.1080/25765299.2018.1449346.
Milenkovic, A., Popovic, B., Dimitrijevic, S. and Stojanovic, N. (2019) Binomial Coefficients and Their Visualization. Proceedings of the Training Conference History of Mathematics in Mathematics Education, Jagodina, 26-30 October 2018, 41-45.
https://scidar.kg.ac.rs/bitstream/123456789/13117/1/
Rosalky, A. (2007) A Simple and Probabilistic Proof of the Binomial Theorem. The American Statistician, 61, 161-162. https://doi.org/10.1198/000313007X188397
Hwang, L.C. (2009) A Simple Proof of the Binomial Theorem Using Differential Calculus. The American Statistician, 63, 43-44.
https://doi.org/10.1198/tast.2009.0009
Yoon, M. and Jeon, Y. (2016) A Study on Binomial Coefficient as an Enriched Learning Topic for the Mathematically Gifted Students. Journal of the Korean School Mathematics Society, 19, 291-308.
https://www.koreascience.or.kr/article/JAKO201630932328637.page
Ferreira, L.D. (2010) Integer Binomial Plan: A Generalization on Factorials and Binomial Coefficients. Journal of Mathematics Research, 2, 18-35.
https://doi.org/10.5539/jmr.v2n3p18
Norton, A. (2000) Student Conjectures in Geometry. 24th Conference of the International Group for the Psychology of Mathematics Education, Hiroshima, July 2000, 23-27.
Nurhasanah, F., Kusumah, Y.S. and Sabandar, J. (2017) Concept of Triangle: Examples of Mathematical Abstraction in Two Different Contexts. International Journal on Emerging Mathematics Education, 1, 53-70.
https://doi.org/10.12928/ijeme.v1i1.5782
Astawa, I., Budayasa, I.K. and Juniati, D. (2018) The Process of Student Cognition in Constructing Mathematical Conjecture. Journal on Mathematics Education, 9, 15-26. https://doi.org/10.22342/jme.9.1.4278.15-26
Stefanowicz, A. (2014) Proofs and Mathematical Reasoning. Mathematics Support Centre, University of Birmingham, Birmingham.
https://www.birmingham.ac.uk/Documents/college-eps/college/stem/Student-Summer-Education-Internships/Proof-and-Reasoning.pdf.