Triple Reflections- A Discourse on twin Prime Conjecture, Pascal’s Triangle, and Euler’s E

I am not a trained mathematician as I read Economics, Geography and Statistics at undergraduate level at the university. However, fro...

I am not a trained mathematician as I read Economics, Geography and Statistics at undergraduate level at the university. However, from infancy I knew I had the flair for mathematics. The reason I could not read mathematics at the university was that coming from a humble and under-privileged background in Ghana, my parents could not send me in the mid-60s to a secondary school because of poverty.

 Despite passing the secondary qualifying exams with flying colours, I could not attend a secondary school education because my parents could not afford it and I ended up going to a teacher training college. After teacher training, I did self-tuition and passed all the qualifying O-Level and A-Level exams and got admitted to the prestigious University of Ghana, Legon in Accra.

In my first year, I read Introduction to Mathematics as part of the Statistics course and also did Mathematics for Economists as well as Statistical Methods/Techniques of Geography. I remember my mathematics professor then used to praise me a lot in front of my course-mates for my brilliance and he jokingly told the class that for always giving correct answers to his questions, he was going to give his sister tome to marry. After one year, I was selected to read Statistics and Economics for the second year.

Unfortunately, the behaviour of a senior professor in the second year made me change and go to read Geography instead of Statistics. The said professor and Head of Department kept drumming in our ears and threatening us that knowledge of Probability 1 which we had missed in the first year would be assumed in the teaching of Probability 2 in the second year. That went on for three weeks without him teaching us anything concrete so I quit. My colleagues who remained lamented that if I the ‘professor’ was leaving, what were they to do? However, they were determined and they stayed and sailed through to the final year.

In my first ever-published article with Global Journals that is based in Framingham, Massachusetts in the USA, I wrote on the topic, ‘Triple Reflections- A Discourse on Twin Prime Conjecture, Pascal’s Triangle, and Euler’s E’. My article was precipitated by an announcement in 2013 that Yitang Zhang of New Hampshire University in the US had cracked the Twin Prime Conjecture, one of the hardest unsolved mathematical puzzles.  Before that, earlier on in 2012, I had put a small write up on that I had come across some arithmetic terms which prove the infinitude of twin primes.

 At that time, I had stumbled upon the fact that the sum of the reciprocals of the binomial coefficients in Pascal’s triangle summed up to 2. I thought I had made a startling discovery on my own only through further research that one person called Brothers (2012) had published a paper on the same point. Brothers article included how Euler’s E could be derived using ratios formed out of Pascal’s triangle. So, one thing led to another.

I therefore decided to do an article embracing all three concepts of the Twin Prime Conjecture, Pascal’s Triangle, and Euler’s E. Brothers’ article of 2012 showed how the multiplication of the coefficients in Pascal’s triangle could be used to form ratios to derive Euler’s E.  The ratios are Sn+1/Sndivided by Sn/Sn-1 where Sn represents holding a line of coefficients constant and relating it to the products of the coefficients above it (Sn-1) and the one below it (Sn+1).  As this process is continued by expanding the coefficients further, the result of dividing the ratios is that the result approaches the value 2.7183, which is Euler’s E. E is derived by another method by summing the reciprocals of the natural numbers to infinity(1/1+ ½+ 1/3+ ¼ +…….+ 1/n) or   Σ (1+1/n)n

I discovered that when you generate twin primes that have a common difference of two,  I discovered arithmetic number patterns or series of the forms 6n-1 , 6n+ 1, 30n-1, 30n+1, 30n-17, 30n+1 among others.Sebah&Gourdon (2002) had done similar series in France before me. The Twin Prime Conjecture that twin primes are infinite dated back to 300 B.C. during the time of Euclid and Eratosthenes who had developed sieves to isolate the twin primes. My work elaborated in my article had taken similar lines.

 In 1849, Polignac, a French mathematician at the French Academy had posed the Twin Prime Conjecture that there is infinitude of twin primes which some mathematicians doubted since the intervals between them get longer and rare. However, my article proved that notwithstanding that, using probability estimation, twin primes constitute 3.49% of all natural numbers, and therefore they are significant in that since natural numbers go to infinity, twin primes also do go to infinity, invoking the arguments of the Central Limit Theorem in Statistics, Fractal Geometry of Mandelbrot, and Self-Similarity theorem of Feigenbaum.

 My discussion made reference to the works of Ramanujan, Hardy-Littlewood, ViggoBrun, Nicely and others who were connected to the task of finding a solution to the Twin Prime Conundrum.
My approach was not the traditional or complex and orthodox mathematical lemmas and proofs, but rather a method of using critical thinking, logic and number patterns to discern solutions. I think this novel approach may sound simplistic yet it is a novel approach from the Social Science point of view. My article is a gestalt or amalgam of different related concepts which reveal how many mathematical concepts are closely interwoven and related in a holistic manner.

I was  able to show that all the twin prime numbers end with only four digits namely 1, 3, 7, and 9 even though not all numbers that end with these digits are twin prime. Finally, I came up with an equation linking Euler, Pascal and Brun namely, Euler2 – Euler * Pascal = Brun
Where Euler = 2.7183, Pascal = 2, Brun = 1.96. I thought this was innovative to honour the work of these mathematical titans. I ended the article by suggesting the practical ways to produce twin prime numbers and also pointing to the practical uses of twin primes.

In my second article, I attempted a solution to Goldbach’s Conjecture which states that an even number is made up of the sum of two prime numbers. The Goldbach’s Conjecture was first hinted by a German, Christian Goldbach in1749 in a letter tohis countryman and fellow mathematician, Leonhard Euler. Euler acknowledged the letter and said he could not solve the problem. The problem has stood for 275 years unsolved.

 My article in Global Journal Science Frontier Research is entitled, “Splitting Goldbach’s Twin Prime Conjecture Asunder of Even numbers’. My title sounds preposterous but that was not the intention. The title rather refers to the simple method I devised to solve the problem by taking samples of even numbers and splitting them into two and finding all the prime numbers above one half of it and those primes above the other half.

I discerned that only a practical method could solve the conjecture without using elaborate mathematical arguments or derivations. I was able to show how if an even number ended with any of five digits, how to eliminate some of the identified prime numbers and thereafter, subtract the remaining prime numbers from half of the even number. Thereafter, I showed that the remaining numbers will reveal prime conjugates of the even number being dealt with and show allits twin prime conjugates. I used many samples of even numbers and arrived at the conclusion that the larger an even number, the more pairs of twin prime conjugates there will be.
 My results also showed that the solution of the twin prime conjugate is directly linked to the Twin Prime Conjecture of Polignac that he proposed in 1849. I came to the realization that all the seven or so unsolved mathematical problems are linked up in some kind of ways. Past and previous attempts to solve Goldbach’s conjecture include the works of Melfi (1996), Hardy-Littlewood (1966), Wang (1984), Chen (1995), Ramare (1995) Marshall (2017), Zhou (2019), Wu (2007) and Helfgott (2013).
My approach showed that it is true that all even numbers to infinity comprise at least two prime conjugates, and that after those conjugates are found we can use arithmetic terms to generate higher even numbers by adding one to the twin prime numbers that we generate. I also provided a flowchart of how to prove Goldbach’s Conjecture and how to practically use the twin prime conjugates so generated.

Time spent on writing the articles
I worked on and off the first article over a period of eight years. My workload at my workplace was in my way as I taught undergraduate, graduate and professional students in various courses such as Human Resource Management, Economics, Strategic Management, Organisational Behaviour, Managing People Across Borders, Business Communication, Research Methods, Public Finance, and supervising and marking dissertations. Besides I was involved in designing many course outlines and writing Course Modules, developing policy documents, and attending management meetings. I spent only four days to write the second article because I had gained a lot of insight from the first article.

Who inspired me to do this work?
My senior in elementary school went to Imperial College in London and he obtained his PhD in 1979. That showed me that I had been in the wilderness for long by teaching without developing my full potential. Even though I have two masters degrees in Public Administrationand an MBA, I am yet to make time to complete my a PhD. I have lectured and mentored DBA students. My former professors at UNISA and Ghana inspired me a lot including Professor Erbyn, Professor Benneh, Prof Samuel Adu-Gyamfi, Professor Benning, among others. From my hometown Winneba in Ghana came professors in Mathematics, Geography, Theology, Agriculture, Physics and Biology.
Some became Vice-Chancellors and Deans such as Prof B.A. Dadson, Prof Kwamina Dickson, Prof Abbiw Jackson, Prof C.C.T. Blankson, Prof Amoasi, Prof Kwesi Dickson, Prof YanneyEwusi, and Prof Acquaye. All these inspired me as some of them are related to me on my father’s lineage. Not to forget Ghanaians working at NASA such as Dr Isaiah Blankson, Dr Ave Klutse, Dr Trebi-OlennuAshitey, Dr Macgbonlurin, Dr Emeagwali, among others.
Also we have world renowned inventors and scientists suchas Dr Thomas Mensah and Dr Victor Lawrence. The late Secretary-General of the UN, Kofi Annan was my role model. May his soul rest in peace.

Future works?
I have many plots of novels to finish writing, and one unpublished poetry book in manuscript form. I plan to write many more articles on two more unsolved mathematical problems. I want to write books in Geography, Basic Statistics, Management, Communication Skills, English Language, and Ghanaian Culture.

My publication experience with Global Journals
Global Journals are super-efficient as make you feel like a king or queen. They answer email within minutes and hours. They have high profile staff who are highly qualified. They are professional and their Journal is second to none in design. They attract the best of the best in academia. They are also humane in that after my first publication which cost me a lot of money, thanks to a sponsor, they decided to give me a generous discount because some of us receive very low pay and we do not receive research funding for our publications.
In Zambia, we scrape through to get by as we receive the equivalent of one thousand dollars monthly salary which cannot cater for our upkeep, research needs and our dependants.
 I will recommend academicians to publish with Global Journals by forming publication partnerships for joint publication so that they can afford the high charges at Global Journals. Global Journals give you constant updates on progress of your paper and also theyput you at ease and keep prompting you to work hard to improve your work. They give you all the support you need. Their state of the art technology such as the Softinator is a first as this software is able to give a grade to your work in terms of spelling errors, grammatical errors, syntax, among others.
 I have experienced American English at first hand through publishing with Global Journals. American English looks weird tome though but a potential publisher has to conform to their standards. I will advise Global Journals to be flexible by accepting both British English and American English and use them side by side.

My work at my University, ZCAS University
This year marks fifty years of teaching for me. I have taught across the spectrum in Ghana, Nigeria and Nigeria. Sometimes I find it frustrating at work that some people do not recognise your academic track record and they want to belittle you by not giving you the recognition you deserve.
 I continue tomentor and motivate students. Currently one of my former students is doing her PhD at the University of Berlin in Agricultural Economics. Another one has gained admission with full scholarship to Strathclyde University in Scotland. Her brother who was alsomy student has alsobeen offered a scholarship for his PhD at Bradford University in the UK. My former student at the International School of Lusaka did his undergrad and masters in Physics at Harvard University and now a doctorate and Rhodes Scholar at Oxford University. Many ZCAS University students have had academic references written for them by me which have helped them to gain admission to prestigious universities around the world.

 A former student of mine went to MIT, and went on to do JD at Harvard. Others went to Purdue, Brown, Dartmoor, Nottingham, UCT, and King’s College, among others. As Head of Research, I am in charge of vetting article publications by lecturers for putting on ZCAS University Repository and also advising on research collaborations with other institutions. We have done collaborations and still doing so with Zambia Institute of Policy Analysis and Research (ZIPAR). We have signed an MOU with Jesuit Centre for Theological Research (JCTR) to collaborate efforts in conducting socio-economic research on living conditions in Zambia.
Kwesi Atta Sakyi, MBA, MPA, BA, Dip. Bus., Cert. A 4Year, MEAZ (Member of Economic Association of Zambia), MORT (Member Oxford Round Table)

About Author :  Kwesi Atta Sakyi is a Ghanaian Lecturer at ZCAS University in Lusaka.  He has teaching experience in secondary schools  and  colleges in Ghana, Nigeria, and Zambia. He taught Economics and Business Management at the International School of Lusaka before moving on to lecture at the Zambia Centre for Accountancy Studies to undergraduate and graduate students in Strategic Management, Managing and Leading People across Borders, and Foundations of Research and Scholarship, among other courses. Sakyi has a BA (Hons) degree in Economics from the University of Ghana, NDP and MPA from UNISA, and recently completed an MBA from UNZA.He has published two poetry books and has journal articleson Google Scholar. In July 2015, he was made a member of the Oxford Round Table where he presented a paper entitled, ‘Early Childhood Education-Penetrating the Impenetrable Issues’. He is an Executive member of the Economics Association of Zambia. His hobbies include photographyand writing.

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Global Journals | Science Innovations & Stories Blog: Triple Reflections- A Discourse on twin Prime Conjecture, Pascal’s Triangle, and Euler’s E
Triple Reflections- A Discourse on twin Prime Conjecture, Pascal’s Triangle, and Euler’s E
Global Journals | Science Innovations & Stories Blog
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