The Marvel of Prime Numbers and Prime Gaps - Article on Mathematics

One may wonder why mathematicians are obsessed with prime numbers and the prime gaps. Prime numbers are popular with mathematicians because they are the building blocks of all numbers. Very large prime numbers are also important in cryptography as they cannot be easily cracked even via the use of a supercomputer (Hoffstein, Pipher & Silverman, 2016). Yitang Zhang is one of the most influential scholars in the science of prime numbers (Thomas, 2014). One of his most important works in this area is Bounded Gap between Primes,' a mathematical research article published in 2014 in Annals of Mathematics at the Princeton University. The work relates two prime conjecture and aims to prove that there exists an infinite number of prime with a prime gap of two.

A prime number refers to a number which can only be divided by 1 and itself without a remainder (Zegarelli, 2015). The prime numbers between 0 and 10, for instance, include 2, 3, 5, and 7. Prime numbers are the basis of all numbers, and any positive integer greater than 1 is either a prime number or a product of prime numbers. A prime gap refers to the numerical difference between two successive prime numbers (Zegarelli, 2015). The prime gap between 5 and 7, for example, is 7-5=2. Other prime numbers with a prime gap of 2 include 59 and 61 as well as 1,000,000,007 and 1,000,000,009.

Yitang Zhang (2014) published the proof of a theorem which many mathematicians have considered as an initial step towards the solution of the twin primes conjecture: a longstanding and unsolved problem in the number theory which questions how far apart prime numbers can be. In the number theory, twin primes conjecture refers to a theory by Euclid that states that there are infinite numbers of prime numbers whose prime gap is 2 (Klarreich, 2014). Why, then, is solving the two primes conjecture important? It will enable mathematicians to predict prime numbers and the gap between successive prime numbers without using large computing power. The conjecture has not yet been proven or disproven. At the moment, the largest prime numbers depicting a prime gap of 2 consists of 200,700 digits each. Euclid also proved that the number of prime numbers is infinite. What about the prime gap then?

Mathematically, it has been proven that as one goes along the number line, prime numbers become rare as twin primes become even rarer (Klarreich, 2014). Nevertheless, Zhangs key finding was that even as prime numbers become rare as one moves along the number line, there are usually prime pairs bounded apart within 70,000,000. From Zhangs theory, no matter how big a prime number is, there is another within a range of seventy million. This may seem enormous, but to mathematicians, it is a very small number when compared to infinity. Considering that there are proven prime numbers consisting of more than a hundred thousand digits, the importance of Zhangs discovery is phenomenal.

Zhang used a combination of existing mathematical formulae to develop his theorem. He realized that the prime gap between any successive prime numbers, except 2 and 3, lies in a range between 2 and H. He then worked out to find the value of H. Using Dickson-Hardy-Littlewood conjecture, Zhang assumed that in a large group of k whole numbers, there exist at least two prime numbers. What is the value of k? To find the value of k, Zhang used a variety of mathematical formulae and got a variety of answers. To accumulate all the answers he took the maximum value of k and using the formula, he settled with 70,000,000 as his value of H.

Apparently, there have existed some questions on prime numbers that have puzzled mathematicians for a very long time; for example, what is the gap between two successive prime numbers? Are these gaps small or large? How often do these gaps appear? Zhang was the first mathematician to demonstrate that gaps between successive prime numbers do not increase indefinitely as prime numbers increase to infinity. From his work, it is vivid that for every prime number, there is another prime in its proximate in a range of 70,000,000. His findings were more impressive as Zhang was a relatively unknown scientist and the twin prime conjecture a hopeless' part of pure mathematics due to lack of a conditioned approach in which the problem can be tackled (Thomas, 2014).

In summary, although Zhang did not solve the two primes conjecture, he proved that even the most difficult and seemingly unworkable mathematical problem can be solved. After Zhangs discovery, hundreds of scientists worked on the two primes conjecture using his method and reduced the gap to 246. If this bound is reduced to 2, it will represent the solution of the two primes conjecture.

References

Hoffstein, J., Pipher, J., & Silverman, J. (2016). An introduction to mathematical cryptography. New York, NY: Springer New York.

Klarreich, E. (2014). Unknown mathematician solves elusive property of prime numbers. WIRED. Retrieved September 8, 2017, from https://www.wired.com/2013/05/twin-primes/

Thomas, K. (2014). Yitang Zhangs Spectacular Mathematical Journey. Institute for Advanced Study. Retrieved September 8, 2017, from https://www.ias.edu/ideas/2014/zhang-breakthrough

Zegarelli, M. (2015). Basic math & pre-algebra. Hoboken, N.J: J. Wiley & Sons.

Zhang, Y. (2014). Bounded gaps between primes. Annals of Mathematics, 179(3), 1121-1174. http://dx.doi.org/10.4007/annals.2014.179.3.7