Problem of the Day #10

Hooray, we are on 10!
So, more patterns, oh, and this has something to do with twin primes.

What comes next in this sequence? 4, 6, 12, 18, 30, 42, 60, 72, 102, 108,...?
Have fun solving!

Twin Primes (Conjecture)

First, what are twin primes?

Prime numbers are numbers that have only two factors, itself and one.  

Twin primes are two prime numbers that have a gap of two. For example, 3 and 5. The twin primes from   1-100:

As you can see, the gap between twin primes becomes larger and larger. This pattern continues as the numbers get larger. The primes are more infrequent.

The twin prime conjecture states that there are an infinite number a twin primes. Although it seems obvious, like the Goldbach conjecture, it is very hard to prove.
An advancement was made by a Chinese mathematician who was previously unknown. He turned in a paper proving that the distance between primes is not infinity but less than 70 million. Yitang Zhang was what seemed like, a retired mathematician who was far from all the action. He used work from predecessors in advancement of knowledge of primes to help his own work.
Mathematicians are working on narrowing the gap down to two.

Next time: the ABC conjecture. 

Goldbach Conjecture - Unproven

The original statement Goldbach made was: Every integer greater than two can be written as the sum of two prime numbers, he wrote about this to a fellow mathematician called Euler. However, Goldbach considered 1 a prime number which we don't today. If he had thought that up today, it would be: Every integer greater than five can be written as the sum of three prime numbers. This conjecture has only been tested up to 400,000,000,000,000. Since numbers go on forever, it is hard to prove the conjecture.

Euler became interested in this problem and answered with an equivalent version: Every even number larger than two can be written as the sum of two prime numbers.

There is also the weak Goldbach conjecture. This conjecture states that every odd number larger than 5 can be written as the sum of two prime numbers.

Many mathematicians have tried to prove this. There is a $1,000,000 prize for anyone who can solve this from March 20, 2000 to March 20, 2002 as an encouragement to young mathematicians but the prize remained unclaimed. This conjecture is, like many things in math, hard to prove. Like many things in math, it is easy to understand yet frustratingly hard to prove. No one really knows why it is so hard to prove. 

Schnirelman proved that every even number is a sum of not more than 300,000 primes. This is however, very far from merely two primes! Estermann proved that most even numbers can be written as the sum of two prime numbers.

As for the weak Goldbach conjecture, Vinogradov proved that a sufficiently odd number can be written as the sum of three prime numbers.

Chen also proved that a sufficiently large even numbers are the sum of a prime and the product of two primes.

This link that will help you understand more. 

Next Time: Twin Primes