International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 11 Issue: 10 | Oct 2024
p-ISSN: 2395-0072
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On Extending Goldbach’s Conjecture: The Patterns of Isolated and Twin Primes into The Even Sums Rudranil Sahu1 1Department of Mathematics, Burdwan Raj College, The University of Burdwan
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Abstract - Goldbach’s conjecture states that every even
claim its validity for all the even integers. In this paper, we have concentrated on the patterns of twins and isolated primes into even sums. Here the presented hypothesis connects the even integers with twin prime pairs [7]. Apart from all of them, we are going to explore the patterns of isolated primes together with twin primes in a special case of even integers. This rudiment and ideas regarding primes would play a pivotal role in the advancement of Goldbach's conjecture.
integer (greater or equal to four) can be expressed as the sum of two primes. Mathematicians have tried to prove it analytically, from the centuries. Unfortunately, all such attempts couldn’t prove it mathematically. We have utilized the patterns of even sums to extend the actual conjecture. In this study, we've extended the conjecture by imparting two hypotheses. Our objective relies upon the necessity of ‘unveiling the isolated and twin prime pattern’ within even integers. In this article, we have illustrated two hypotheses, based on our intriguing observations. The former insists that, in the sum of two primes that represent an even integer (greater or equal to six), at least one of these primes must belong to a twin prime pair. Due to further investigation, we have presented the latter, which strengthened the former. It says that there exists a subset of the set {E= 12m: m is positive integer}, where each member can be expressible simultaneously as the sum of two primes (both are part of a twin prime), the sum of two primes (both are part of the same twin prime pair), the sum of two primes (one twin prime, and isolated prime), and the sum of two primes (both are isolated primes). To support our claim, we’ve performed extensive computational verification using Python software. The former is verified up to 250000 and the latter confirms its validity up to 5000. We expect our future works on their larger verification will make a new avenue in the course of number theory.
2. Essential preliminaries: For general readers, we discuss here some basic elementary concepts, which are relevant to our work.
2.1 Primes: A prime number cannot be represented by a product of two smaller natural numbers than itself. Those positive integers are divisible by a unit and itself only, known as ‘prime’. There are infinitely many primes, as established by Euclid around 300 BC. No known simple formula separates prime numbers from composite numbers except their basic property 2.2 Twin prime pairs: The prime numbers are defined to be worthy individually but when we refer to a ‘twin prime’ then it is to be understood they exist in pairs. A twin prime is a pair of numbers difference of which is two. Sometimes the term ‘twin prime’ is called ‘twin prime pair. There are a few examples of twin prime pairs as (3, 5), (5, 7), (11, 13), (17, 19) etc.
Key Words: Prime numbers, Twin prime pairs, Isolated primes, Goldbach Conjecture
2.3 Isolated primes: Isolated primes are such primes, which has no twin within the gap of two. The non-twin primes can be categorized as isolated primes. There are some isolated primes given as 23, 37, 47, 53,67, 79, 83, 89, 97, etc.
1. INTRODUCTION The history of number theory is a rich tapestry woven across millennia, showcasing humanity’s enduring fascination with the properties and patterns of numbers. The journey begins with ancient civilizations like Babylonia and Egypt, where mathematical insights were first recorded [10]. The Goldbach conjecture is one of the intriguing conjectures in “Number Theory”. It states that every even number (greater or equal to four) can be represented as the sum of two prime numbers [9]. The claim is satisfied up to a very massive range. Mathematicians are still trying to prove it for countless even integers [8]. Proving the converse is an easy task and could be done through elementary learning. Unfortunately, till now, no such analytical proof exists to
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2.4 Goldbach’s conjecture: In 1742, Christian Goldbach proposed an interesting hypothesis in a letter to Leonhard Euler. Later it is known to us by the name ‘Goldbach’s conjecture’. This simply relates even numbers with the primes [4]. Mathematicians have wrestled with this conundrum for centuries to prove it analytically. It states that every even number greater than 4 can be expressed as the sum of two prime numbers.
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