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!

yitang zhang

james maynard

Unsolved Problems In Mathematics!

Ever wondered “Math is complete or not?” If you have thought Math is Complete than in some cases you may be wrong. Wrong in the sense that there are still some problems in Mathematics, which are still remains to be solved or we can say still they are unsolved. Yeah you are getting it right. So, what are the problems they are still remains to be solved?

A few of them are as follows:

1. The Riemann’s Hypothesis

2. The P vs. NP problem

3. The Navier-Stokes Equation

4. The Hodge Conjecture

5. Yang-Mills Theory and mass Gap Hypothesis

To solve these problems, you need a good Mathematical background. And also, in 2000 “The Clay Mathematics Institute” situated in Peterborough, New Hampshire, United States has announced $1 million prize to the person who ever solve any one of these problems.

We are not going to talk about any of these problems but there is a problem in Mathematics which can be understood by any sixth-grade student and still remains to be solved. And the problem is named as “Collatz Conjecture” or sometimes known as “3n+1” problem.

Collatz conjecture:

This problem or we can simply say Conjecture is named upon German Mathematician Lothar Collatz (July6, 1910- Sep. 26, 1990). It states that every whole number either even or odd eventually goes down to 1. And for this we have to consider the following operation on any positive Integer n;

1. If the number is even divide it by two that is

(n/2)

2. If n is odd triple it and add one that is

(3n+1)

In Arithmetic notation we define a function f as follow:

The sequence of number involved is known as Hailstone sequence or Hailstone numbers (because the value usually goes up and down like hailstone in cloud.) or as wondrous number. People have tried this function for the number up to 2⁶⁰ and that still get back to 1.

Now the question may arise what is the problem on this function, we will talk about it on the last first let’s see some illustrative examples:

For example, take no. 12

12 is even so divide it by 2.

12/2=6 (even)

6/2=3 (odd)

(3*3)+1=10 (even)

10/2=5 (Odd)

(5*3)+1=16 (even)

16/2=8 (even)

8/2=4 (even)

4/2=2 (even)

2/2=1

Here again if you think 1 is odd, then we can do it again for 1

(1*3)+1=4 (even)

4/2=2 (even)

2/2=1

Back to 1 again, so the cycle

4 2 1

This cycle keeps repeating after we got to 1.

Now take a number less than 12, take 9 (smaller than 12)

Back to 1 again, here we can see that 12 greater than 9 took more step to become 1. But also if we take 8 there will be more less steps.

Taking example of those numbers which will take more steps to descend to 1:

1. Between 1 and 100:

· 27 take 111 steps to become 1; it climbs up to 9,232 before descending to 1.

· 54 and 55 both takes 112 steps to descend to 1; both of them also climb up to 9,232.

· 97 take 118 steps to descend to 1.

2. Between 1 and 1000:

· 703 take 170 steps to descend to 1 it climbs up to 250,504.

· 937 take 173 steps to descend to 1 it also climbs up to 250,504.

· 871 take 178 steps to descend to 1 it climbs up to 190996.

3. Between 1,000 and 10,000 is 6,171 which take 261 steps to descend to 1, It climbs up to 975,400.

4. Between 10,000 and 100,000 is 77,031 it take 350 steps to descend to 1, it climbs up to 21,933,016.

5. Between 100,000 and 1 million is 837,799 it takes 524 steps.

6. Between 1 million and 10 million is 8,400,511 it takes 685 steps.

7. Between 10 billion and 100 billion is 75,128,138,247 it takes 1,228 steps.

Actually, the problem in this Conjecture is that there is no certain rule that, the larger number will take more steps or smaller number will take fewer steps to become 1. For example, the number 909 take 15 steps to descend to 1 where as the number 7 take 16 steps. There are so many other examples like this. Also, we cannot find any pattern of numbers descending to 1.

So, it’s up to you!

References:
Devlin, The millennium problems
Wikipedia, Numerphile