Primes and Twin Prime Conjecture

Mathematics is one of the most interesting part of science. Usually Math deals with study of patterns that is, number theory is the study of patterns of numbers, geometry is study of pattern of shapes, algebra is study of pattern of putting things together, trigonometry is measurement of shapes, calculus is patterns of continuous motion and change, probability theory is pattern of repetition in random events, statistical theory is pattern of real-world data, and logic is pattern of abstract reasoning.

Primes

What are Primes? Are they really important part of mathematics?

Yes, primes are really important part of mathematics and are also known as building block of mathematics. A question may arise why primes are building block instead of natural number or real number? The answer is that every integer greater than 1 is either prime or can be expressed as product of prime number. That is every number you see around is either prime or is product of prime that’s why primes are building block of mathematics as elements are building blocks of chemistry.

For example, let us take random numbers: (To show every number is unique decomposition of primes)

1200=2*2*2*2*3*5*5

28=2*2*7

Euclid

The concept of primes is that “A number greater than one is said to be prime if it is divisible by one and number itself.” A few examples of prime numbers are 2,3,5,7,11 and so on. The history of prime shows earliest surviving records of explicit study of prime number comes from Ancient Greek Mathematics.

Among the small number’s primes are very common. Of the numbers 2 to 20, the numbers are 2,3,5,7,11,13,17,19 are prime a total of eight out of nineteen. The remaining numbers are all composite; that is, they are not prime, since each number can be evenly divided by some smaller numbers (apart from 1). As we look at larger and larger numbers the primes appear to be thin out. While there are 5 primes below 10, there are only 24 below 100 and just 168 below 1000. If we see the average rate at which prime appears, they have an average rate of 0.5 below 10, 0.24 below 100 and just 0.168 below 1000. The table of prime and average up to 1 million is:

N	10	100	1000	10000	100000	100000
Average	0.5	0.24	0.168	0.123	0.096	0.078

The farther we go the smaller the average becomes. A few questions that arises over here are;

· Does this thinning continue?

· Or do we reach a point where it reverses and we find a lot of primes?

· Or do we reach a point where we do not find primes at all?

These questions were answered by Euclid around 300 B.C. He proved that the primes continue forever and there are infinitely many of them, which is also known as Euclid’s Theorem.

Few facts about Primes:

1. Prime numbers have only two factors 1 and itself.

2. The only even number that is prime is 2.

3. There are infinitely many prime numbers.

4. Every number can be uniquely expressed as product of prime numbers.

5. The prime numbers cannot be formed by product of two natural number that are both smaller than it.

6. No even number greater than 2 can be prime as it can be written as the product of 2*n/2.

7. The largest prime found till date has 23,249,425 digits.

Twin prime Conjecture:

Conjecture in mathematics is a specific statement that is thought to be true but has not been proven yet. Its similar to hypothesis. The main difference between hypothesis and conjecture is that every hypothesis can be tested but not every conjecture can be tested. A conjecture can become first a hypothesis then a theory then at last a law. So, there are many Conjectures in mathematics which are unsolved and there are conjectures also which has been solved (Poincare Conjecture is one of the famous Solved conjecture). Twin prime is also one of the unsolved Conjecture in Mathematics.

Twin prime is a prime number that is either two less or greater than another prime. In other words, twin prime is a prime that has a prime gap of two. for example, member of the twin primes are 17 and 19 The first few twin primes are:

(3,5), (5,7), (11,13), (1719,), (41,43), (59,61), (71,73), (101,103), (107,109), (137,139), … Usually (2,3) is not considered as twin primes.

The Questions of Twin Prime Conjecture are:

1.) Are there infinitely many twin primes?

2.) If there are finitely many, which is the largest twin prime?

The work of Yitang Zhang in 2013, as well as work by James Maynard and others has made substantial progress towards proving there are infinitely many twin primes but still it remains to Solved!

This is an open Question that has remained unsolved, So its up to you also!