# Twin-Prime Conjecture Research (Term Paper Sample)

Research on Twin-prime conjecture with a focus on introduction highlighing on the history and application, differential and integral calculus, then finish with a conclusion

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Twin-Prime Conjecture

1 Introduction

Polignac’s conjecture is another phrase used when referring to twin-prime conjecture and it is studied in number theory. Now, to specifically define it, the twin prime conjecture is associated with having prime numbers that appear to be apart by two steps (Letzter), meaning that the difference from the former and present is 2 and they are infinitely many of these in the number line, however, as the numbers become large, they start being rare (Hosch). For example, 3 and 5, 269 and 271, and 599 and 601.

* History

* It was Alphonse de Polignac, a mathematician from France who came up with the first statement concerning the twin-prime conjecture in 1846 (Hosch). In his description, it detailed the infinite expression of the difference associated with two prime numbers that consecutively follow each other on the number line (Hosch). Additionally, his explanation showed that there was an even number; 2 derived from the difference between the two consecutive primes (Hosch). For example, 2 = 5 – 3 = 19 – 17 = 31 – 29 = …

* On the other hand, Euclid is credited with providing the first-ever existing proof known to be the oldest as far as twin prime conjecture is concerned, which is referred to as Euclid's twin-prime conjecture (Hosch).

* In 1919, Viggo Brun, a mathematician from Norway made progress in the twin-prime conjecture whereby he came up with the summation known as the Brun’s constant that is derived from getting the twin-primes’ reciprocal (Hosch). Specifically, the Brun’s constant is where the twin-primes converge to a sum (Hosch).

* Later on, in 1976, calculations on Brun’s constant was approximated as 1.90216054 (Hosch). The calculation followed a total of primes adding up to 100 billion on the number line (Hosch).

* Thomas Nicely, an American mathematician in 1994 discovered that by using the Pentium chip, newly introduced in the market in personal computers by the Intel Corporation, had flaws in the calculation of Brun’s constant. In 2010, he came up with Brun’s constant where he concluded its value as 1.902160583209±0.000000000781. The value resulted from calculating for Brun’s constant from all primes not more than 2×1016 (Hosch).

* In the same spirit of Nicely, Daniel Goldston; a fellow American mathematician together with Cem Yildirim a mathematician from Turkey published the paper, "Small Gaps Between Primes, " whereby according to their findings, it was established that there exists an infinite number of prime pairs inside a small difference (Hosch). However, there were flaws with their proof and it led to its correction in 2005 with the help of a mathematician from Hungary known as Janos Pintz (Hosch). Eight years later in 2013, Yitang Zhang, an American mathematician showed that in the absence of any assumptions, the difference in the infinite number was 70 million (Hosch). It was specifically done by building on the work of Daniel Goldston, Cem Yildirim, and Janos Pintz (Hosch). Nevertheless, in 2014, the bound of infinite number differing was improved to 246 (Hosch).

* Applications

From existing research, twin-prime conjecture is applied in the answering of questions associated with numbers. Besides that, there is no single specific field or discipline which are known to extensively define real-life issues, but mathematicians enjoy coming up with proofs that pertain numbers.

2 Characteristics of twin-prime numbers

The division algorithm follows that when dividing a prime number or any given integer n with another denoted as positive, i, unique integers exist that make up the division complete such that it is a quotient; q, and a remainder; r (Ruiz 7). These when put together produce the following results n=qi+r, provided that i>r≥0 (Ruiz 7). The equivalence is of importance in this respect because there are other primes that may contradict the difference of 2 between them.

Therefore, to be more specific with details about the twin prime conjecture, below are characteristics that define it detail;

* When there exists any even number greater than 2, its sum is that of two primes (Nd 2).

* When there exist two perfect squares on a number line that are consecutive on the number line, there is a prime number there as well (Nd 2).

* The twin prime number is infinitely many on the number line (Nd 2).

* When written in the form of n2+1, the primes still are infinitely many (Nd 2).

3 Generalization of twin-prime conjecture

There exist two generalizations, namely: the first and second. In the first generalization, it includes a sequence developed by Sophie Germain whereby the elements thereof consist of all twin primes and t no point does the prime factors exceed 3 (The OIES Foundation). It implies that it includes all prime numbers that start from 3, 5, 9,…, 1049 (The OIES Foundation). It is a long list but provides all the important information about the first generalization and the specific elements to understand in the number line (The OIES Foundation). Therefore, it is concluded that when given a proper divisor d of n, the resulting n+d+1 will be a prime number.

On the other hand, the second generalization includes a different process of identifying the prime numbers from the first one (The OIES Foundation). Here, there is a prime number p, which is referred to as n-isolated when there exists no other prime number within the provided interval. For example, given a(n)=1, taking the smallest p; n-isolated prime number and then dividing it by n, the result if neither 2 nor 3 (The OIES Foundation). Besides, as n approaches infinity, dividing the prime numbers by n implies that there is a tendency of it also approaching infinity (The OIES Foundation). Therefore, the second generalization dwells on the n-isolated prime number and whether the difference between two primes when divided by n results to either 2 or 3 to help in determining the general understanding of the twin-prime numbers.

4 Sieve Method

The use of Sieve theory in number theory is an important proof in showing the existence of infinitely plenty of twin-prime numbers (Nd 6). Although sophisticated, it is critical to mathematicians considering the fact that its use spans over 52 years (Nd 6). From the Twin Prime Problems together with Goldbach’s Conjecture, different questions existed with no answers and this is the method that helped in uncovering some of the unexploited areas (Nd 6). The method is known arithmetically sieve prime numbers on grounds that the provided integer sequence is finite, the prime numbers follow the desired sequence, and the existing number z(≥2) (Nd 6-7). We then proceed to separate the sift and unsifted elements where the unsifted has few prime divisors compared to the sifted (Nd 7). In its entirety, the unsifted elements are important for the sieve method as they are used in the estimation of S (Nd 7). However, as far as the method is concerned, there are some difficulties related to finding a general understanding of the primes.

Moreover, as discussed in the history section, Viggo Brun was the first mathematician to successfully apply the sieve method and come up with a proper explanation of the twin-prime conjecture properties (Nd 7). The technique used was hard for mastering by some of the mathematicians and other experts at the time considering the extensity of its full potential hence very few managed to keep that knowledge to themselves (Nd 7). Therefore, it is correct to conclude that the sieve method is used in the estimation of elements from the sifting procedure and there is no instance where many primes exist.

5 Brun’s Constant

In the first 30 years of the twentieth century, there were a number of major conjectures established by mathematicians (Nd 9). It also included theorems that define today’s mathematical calculations in relation to prime numbers or generally, numbers (Nd 9). In light of these, Brun had his work well-received b the mathematicians across the world because his

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