New Discovery in Number Theory

People have been interested in a prime number since the ancient times. Euclid proved that an amount of prime numbers is infinite around 300 BC. Eratosthenes used his algorithm to quickly find huge prime numbers around 240 BC.
One of the most important mathematical discoveries that happened in recent years was likewise related to prime numbers. Yitang Zhang, professor at University of New Hampshire, discovered the maximum difference between two consecutive prime numbers in 2013. He found that the interval cannot be more than 70 million. In his proof he used previous works of Goldston, Pintz and Yıldırım. They proved that there exists some finite gap between primes in 2005. However, they could not identify its value.


To make explanation clearer, I will start with the explanation of a prime number.
D. Underwood states that "a natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive divisors, 1 and the number itself." For example, number "3" is a prime because it can be divided only by "1" and "3". The first twenty-five prime numbers are shown on the Figure 1.

Figure 1. Prime numbers. Link

 The following video explains the prime numbers in more detailed way.

Video 1. Explanation of the Prime Numbers. Link      

As it can be seen from the Figure 1, the average difference between two consecutive prime numbers increases as the prime numbers increase. Using that observation, Dr. Zhang showed in his proof that the gap will not exceed 70 million. Figure 2 illustrates that relation. Yitang Zhang proved the hypothesis based on the principles of number theory. For reference, Ireland and Rosen, who got PhD degrees from Johns Hopkins University and Princeton University respectively, define number theory as a study of natural numbers (i.e. 1, 2, 3, 4, 5, 6, etc.). Actually, the whole proof is 56 pages long and is very complicated. Video 2 explains the proof in simpler way.

Figure 2. Demonstration of Dr. Zhang`s Proof. Link

Video 2. The simpler explanation of Dr. Zhang`s Proof. Link

Dr. Zhang`s proof is the most significant discovery in mathematics since 2009, when Ngo Bao Chau proved the fundamental proposition. As a result, this proof can make huge contribution in science of mathematics. For example, Yitang Zhang`s proof is the starting point in proving twin prime conjecture. For reference, twin primes are two consecutive prime numbers, those difference is equal to "2". For example, "3" and "5" or "11" and "13" are twin primes. Twin primes conjecture states that there is an infinite number of twin primes. The highest known pair of twin primes is "3,756,801,695,685*2^666,689+1" and "3,756,801,695,685*2^666,689-1". Figure 3 illustrates this hypothesis.

Figure 3. Demonstration of Twin Prime Conjecture. Link

In addition, Dr. Zhang`s calculations can be used to prove other hypotheses related to prime numbers, such as Goldbach's conjectures. Goldbach's conjecture is one of the most famous and oldest unproven number theory`s hypothesis. It states that "every even integer greater than 2 can be expressed as the sum of two primes." 

In conclusion, Dr. Zhang`s discovery can inspire more students to study mathematics. This discovery, which was made in the University of New Hampshire, shows that important findings can be made not only in world-famous universities, such as Harvard University, Massachusetts Institute of Technology (MIT), or University of Oxford. 

Additional links for further research:

About Yitang Zhang. Link

About Number Theory. Link 

About Prime Numbers. Link

About Prime Number Theories. Link

Reference List:

"Bounded gaps between primes." Annals of Mathematics. http://annals.math.princeton.edu/2014/179-3/p07 (accessed October 28, 2014).

Dudley, Underwood. Elementary number theory. 1969. Reprint, San Francisco: W.H. Freeman, 1978.

Ireland, Kenneth F., and Michael I. Rosen. "Unique Factorization ." In A classical introduction to modern number theory. 2nd ed. New York: Springer-Verlag, 1990. 1.

Numberphile. "Gaps between Primes - Numberphile." YouTube. http://www.youtube.com/watch?v=vkMXdShDdtY&src_vid=D4_sNKoO-RA&feature=iv&annotation_id=annotation_667840 (accessed October 20, 2014).

Numberphile. "Gaps between Primes (extra footage) - Numberphile." YouTube. http://www.youtube.com/watch?v=D4_sNKoO-RA (accessed October 20, 2014).

"Prime numbers." Khan Academy. https://www.khanacademy.org/math/cc-fourth-grade-math/cc-4th-fact-mult-topic/cc-4th-prime-composite/v/prime-numbers?v=mIStB5X4U8M (accessed October 20, 2014).

Soundararajan, K.. "Small gaps between prime numbers: The work of Goldston-Pintz-Yıldırım." Bulletin of the American Mathematical Society 44, no. 01 (2007): 1-18. http://dx.doi.org/10.1090/S0273-0979-06-01142-6 (accessed October 28, 2014).

"UNH Mathematician’s Proof Is Breakthrough Toward Centuries-Old Problem." University of New Hampshire. http://www.unh.edu/news/releases/2013/may/bp16zhang.cfm (accessed October 20, 2014).

"UNH Mathematician Zhang Is 2014 MacArthur Fellow." University of New Hampshire. http://www.unh.edu/news/releases/2014/09/bp17zhang.cfm (accessed October 20, 2014).

Weisstein, Eric W., "Euclid's Theorem", MathWorld. http://mathworld.wolfram.com/EuclidsTheorems.html (accessed October 20, 2014).

Weisstein, Eric W., "Goldbach Number", MathWorld. http://mathworld.wolfram.com/GoldbachNumber.html (accessed October 20, 2014).

Wikimedia Foundation. "Timeline of mathematics." Wikipedia. http://en.wikipedia.org/wiki/Timeline_of_mathematics#21st_century (accessed October 20, 2014).

Wikimedia Foundation. "Twin prime." Wikipedia. http://en.wikipedia.org/wiki/Twin_prime#Largest_known_twin_prime_pair (accessed October 20, 2014).

Wikimedia Foundation. "Twin Prime Conjecture." Simple English Wikipedia. http://simple.wikipedia.org/wiki/Twin_Prime_Conjecture (accessed October 20, 2014).