Browsing by Author "Kamalu, Chukwunyere"

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    An Application of the Hardy-Ramanujan Theorem
    (2022-02-11) Kamalu, Chukwunyere
    This paper attempts to answer the riddle of why Ω(m), the sum of the exponents of the prime decomposition of m = (2x-2)(2x-3)(2x-5) …(2x-qb) [where qb is the bth prime and b = π(x), the number of primes not exceeding x], always appears to lie in the region of 2π(x), where π(x) is the number of primes not exceeding x. It turns out that an application of the Hardy Ramanujan Theorem shows Ω(m) is almost always exceeded by 2π(x), which has the logical implication the Goldbach’s Conjecture is “almost always true”.
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    Earth, Water & Justice: A Note by the Society of African Earth Scientists on the Environmental effects of Land Grabbing
    (2023-05-02) Kamalu, Chukwunyere
    This brief note inspired by the reports of other agencies, aims to define land grabbing as pertains to Africa and outline the scientific and ethical case against African governments allowing land grab that threatens soil fertility, water and food security. The paper also highlights the weakness of the CDM in providing a loophole to avoid direct reduction of emissions and leading to increased land grabbing. The note concludes with recommendations urging African government to exercise due diligence in protecting the rights and best interests of their citizens, and dissuading them from the danger of forfeiting the lands and birthright of current and future African generations.
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    On Primes of the Form 2x-q (where q is a prime less than or equal to x) and the Product of the Distinct Prime Divisors of an Integer (Revised): A Function Approach to Proving the Goldbach Conjecture by Mathematical Induction
    (2021-02-21) Kamalu, Chukwunyere
    This paper is really an attempt to solve the age-old problem of the Goldbach Conjecture, by restating it in terms of primes of the form 2x-q (where q is a prime less than or equal to x). Restating the problem merely requires us to ask the question: Does a prime of form 2x-q lie in the interval [x, 2x]? We begin by introducing the product, m, of numbers of the form 2x-q. Using the geometric series, an upper bound is estimated for the function m. Next, we prove a theorem that states every even number, 2x, that violates Goldbach’s Conjecture must satisfy an inequality involving a simple multiplicative function defined as the product, ρ(m), of the distinct prime divisors of m. A proof of the Goldbach Conjecture is then evident by contradiction as a corollary to the proof of the inequality.
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    The Ishango Bone: The World's First Known Mathematical Sieve and Table of the Small Prime Numbers
    (2021-02-23) Kamalu, Chukwunyere
    This paper aims to show that the Ishango bone, one of two bones discovered in the 1950s buried in ash on the banks of Lake Edward in Democratic Republic of Congo (formerly Zaire), after a nearby volcanic eruption, is the world's first known mathematical sieve and table of the small prime numbers. The bone is dated approximately 20,000 BC. Key to the demonstration of the sieve is the contention that the ancient Stone Age mathematicians of Ishango in Central Africa conceived of doubling or multiplication by 2 in a more primitive mode than modern Computer Age humans, as the process of "copying" of a singular record (that is, a mark created by a stone tool as encountered in Stone Age people's daily experience). Similarly, the doubling of any number was, by logical extension, a process of copying of any number of records (marks) denoting an integer, thereby doubling the exhibited number (marks). Some evidence for this process of "copying" and thus representing numbers as consisting of "copies" of other numbers, is displayed on the bone and can still be found to exist in the number systems of modern Africans in the region. Unlike previous speculations on the use of the bone tool by other studies, the ancient method of sieving of the small primes suggested here is notable for unifying (making use and explanation of) all columns of the Ishango bone; whilst all numbers exhibited form an essential part of the primitive mathematical sieve described. Furthermore, it is stated that the middle column (M) of the bone inscriptions houses the calculations of the Ishango Sieve. All numbers deduced in the middle calculation column relate to a process of elimination of the non-prime numbers from the sequence of numbers 1,2,3,4,5,6,7,8,9,10 (although numbers 1 and 2 are omitted). The act of elimination is proven by the display of the numbers deduced in the middle column; namely: 4, 6, 8, 9, and 10 and the subsequent omission of these same numbers from the following list leaving only: 5, 7 at the bottom of column M. This elimination process described above is repeated to obtain the primes 11,13,17,19 when eliminating non-primes from the sequence 11,12,13,14,15,16,17,18,19,20. However, only calculations for the sequence 1 to 10 (for numbers above 2) are displayed in column M; as if to exemplify the Ishango Sieve method for the benefit of posterity.