Abstract
The twin prime conjecture has attracted a lot of attention worldwide.It is still an unresolved problem, even though the work of Yitang Zhang has partially resolved it. The author of this paper aims to contribute to the discourse by employing basic mathematics and logic to arrive at some conclusions on the topic, and also to help in breaking new grounds. The researcher used secondary data to build his arguments in an exploratory manner, relying on the existing literature. The paper traces the background of the problem, and points to some of the breakthroughs that were made in the past. The paper examines Pascal’s triangle and, it makes some revealing discoveries on the coefficients. The author also examines Euler’s E, and links it to Pascal’s triangle, and the twin prime problem. Furthermore the author derives new arithmetic terms that he can use to produce infinite numbers of twin primes. The author also discusses how numbers so obtained can thoroughly be checked to be non-composite, thus extending the field of twin primes. The author finally points to the application of twin primes in industry, academia, and other areas of practical knowledge.
Highlights
We can observe from the expansion of binomial expressions of the form (x+y)n that the numerical coefficients of the terms in the expansion can be derived by using Pascal’s triangle, named after the French mathematician(Wong, 2013)
Assert that we can generate infinite series of twin prime numbers using the pair of terms (30n-1) and (30n+1) on the one hand, and on the other side, the dual terms (30n-17) and (30n-19) as well as the other duo terms tabulated above
Twin prime numbers have numerous practical applications in medicine, pharmacology, astronomy, cryptology, data science, and machine learning, among many other uses
Summary
We can observe from the expansion of binomial expressions of the form (x+y)n that the numerical coefficients of the terms in the expansion can be derived by using Pascal’s triangle, named after the French mathematician(Wong, 2013). 1 5 10 10 5 1 These coefficients when, summed up line by line horizontally, form the geometric series to infinity of the form, 1, 2, 4, 8, 16, 32, 64, 128,......, 2n-1. They form the series of the form 2n-1 , where n is an integer from n=1 to n = infinity. If we sum the reciprocals of the series to infinity using a geometric progression, we obtain the sum to infinity as Triple Reflections- A Discourse on Twin Prime Conjecture, Pascal’s Triangle, and Euler’s E a /r (1)
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