You have probably read a news about one professor proving The Prime Gap conjecture. In this post, I will give you an overview of what the excitement is all about in the mathematics community.
This post is written for the high school students and those who are interested in mathematics that are non mathematics majors.
What are Prime Numbers?
Most of us are familiar with prime numbers. A prime number is a positive integer that is divisible only by 1 and itself. The number 5 is a prime number, while 8 is not prime because 8 is divisible by 2 and 4. If we examine the 10 positive integers, it is easy to see that only four are prime numbers: 2, 3, 5 and 7. In the figure below, shown are the prime numbers less than 100.
How many Prime Numbers Are There?
If we explore further, we will notice that as the positive integers get larger, fewer and fewer prime numbers are found. For example, there are 168 primes between 1 and 1000, 135 primes between 1000 and 2000, 127 primes between 2000 and 3000, and 120 primes between 3000 and 4000. Looking at the pattern, one might say that since prime numbers are getting rarer as the numbers get larger, then we have a largest prime number. That is the same as saying that the number of prime numbers is finite (we can count them) — one billion maybe, or some large number.
Well, surprisingly it has been known and proven several hundreds of years ago that there are infinitely many primes. If you are a high school student and plans to be a math major, you will be familiar with the proof during your first year in college.
Some Special Primes
We can also observe from the exploration above that there are primes that are ‘closed’ to each other. The pairs (3,5), (5, 7), and (11, 13) have difference or gap of 2 (can you think of others). These primes of the form (a, b) where b – a is 2 are called Twin Primes. The next question is, how many Twin Primes are there? Well, mathematicians believe that there also infinitely many of them (they call it the Twin Prime Conjecture), but until now no one has shown whether it is true or not.
There are also primes with gap 4 such as (3,7) and (7,11). These primes are called Cousin Primes (Can you believe that?). Again, the question is, are there infinitely many cousin primes? Again, no one was able to show that this is true or false.
Lastly, and I hope you know where there is going to, there are also primes with gap 6 such as (23, 29), and mathematicians call them sexy primes (I am not joking really). Now, are there infinitely many of them? No one knows for now.
Yitang Zhang and the Bounded Gap Conjecture
The questions above are what the latest mathematics hype is all about. Yitang Zhang, a mathematician from the University of New Hampshire has proven that we can find an infinite number of prime pairs (a,b) such that
b – a = N, such that N <= 70,000,000.
What does it mean?
It means that there is a number k less than 70,000 — we do not know k yet, but we know that it is less than 70,000,000, such that
b – a = k
where (a,b) are infinite pairs of primes.
If somebody later proves that k = 2, then the Twin Prime Conjecture is proved.
It also means that if we randomly choose ANY prime number greater than 2, we are sure that if we go 70,000,000 up or down, we can alwys find at least one prime number.