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Long Him Lui The twin prime conjecture is one of the most famous unsolved problems in the field of mathematics known as number theory. Number theory, as the name suggests, deals with numbers, more specifically the relationships between different types of numbers. Some examples are: One of the most notorious groups of numbers that mathematicians deal with are the prime numbers. Prime numbers are tricky to deal with because there is no clear relationship between the nth term and the n+1th term, so we cannot write out a general formula for the set of n primes. The unpredictability and abstractness of prime numbers are precisely what attract mathematicians to the twin prime conjecture, with world-renowned mathematicians such as James Maynard and Terence Tao having contributed significant work to the attempt to solve this mathematical puzzle. Before diving any deeper into the twin prime conjecture, we first need to define the term conjecture. A conjecture is a proposition that is suspected to be true based on preliminary evidence in its favour, but for which no proof or disproof has been found. After a conjecture is proved to be true, it is considered a theorem, becoming a non-self-evident statement that has been proven to be true on the basis of generally accepted axioms (statements that are accepted to be true) or previously established mathematical proofs and other theorems. A twin prime pair is a set of two prime numbers that have a difference of two. Mathematicians typically label differences between prime numbers using the term “prime gap”, so the definition of a twin prime is more commonly heard as: a set of two primes that have a prime gap of two. For example, (41,43) would be considered as a twin prime pair. Logically speaking, twin prime pairs become increasingly hard to detect as we ascend into larger number ranges, but this is to be expected. The general tendency is that gaps between adjacent prime numbers increase as numbers get larger, since there are a larger number of factors that the number can be divided by, resulting in a diminishing chance for a number to be prime. This results in twin primes becoming increasingly rare as we proceed into larger and larger numbers. However, we cannot examine the twin prime conjecture if we cannot prove that infinite primes exist, since the twin prime conjecture suggests that there are infinite twin prime pairs. Even though there is no pattern showing the correlation between all primes, we are sure that there exist an infinite number of primes. The proof of infinite primes can be shown by a proof by contradiction. Our new number q is clearly larger than any prime number, so it is not equivalent to any of them. Since p1 to pn constitutes all prime numbers, q cannot be prime. Since non-prime numbers can be written as a product of primes, it must be divisible by at least one of our finitely many primes pn (where ). However, when we divide q by pn, we obtain a result with a remainder of 1, which contradicts the original assumption that there are a finite number of primes. Therefore, we can say that there exist an infinite number of primes. During the computation of this conjecture, a very common tool that mathematicians use is the function known as the logarithmic integral function (li_k(x)), which has many uses in physics. The reason why this is used in the twin prime conjecture is because the logarithmic integral function is considered a very good approximation for the Prime Counting Function Π(x), which is a function used to calculate the total number of prime numbers in a certain set of natural numbers. The graph shows the total number of primes that exist in a set of N integers. The y axis shows the number of primes and the x axis shows the total number of numbers N. The blue line shows the prime count according to the Prime Number Theorem, while the red line shows the prime count according to the logarithmic integral function. Notice that the approximation is very good for any small to medium sized set of integers from 0 to around 50000, but starts to have a noticeable discrepancy from large sets above 50000 integers. However, a good thing about this estimation is that it is a consistent underprediction, which makes it convenient to adjust for discrepancies. In general, this theorem is able to reciprocate a generalized version of the twin-prime conjecture for any prime gap by taking the average distance between a prime number and a sample space of integers combined with the Prime Number Theorem, which estimates the number of primes within N integers. By taking a product of these two values, we can estimate the average prime gap within N integers.
Lots of work is still being done on the twin prime conjecture. Most recently in 2013, a mathematician called Yitang Zhang, proved that there are an infinite number of prime numbers that differ by 70 million or less. Even though 70 million is still quite far from two, it is considered a huge breakthrough in the mathematical world, and hopefully more research can be completed on this conjecture for it to become one of the most beautiful theorems mathematics has ever seen.
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