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

Math is Golden

You are probably guessing from the title that I'm talking about the Golden ration or phi, the Greek letter. The golden ratio is equal to 1.61803398875 and so on and so forth. More information on phi here. Don't get phi and pi mixed up. Phi is the golden ratio, pi is related to ovaloid things. Got it?
The idea behind the golden ratio is that, if you divide a line into two sections, the longer part divided by the smaller part will equal the whole length divided by the longer part. In this way, you can create a perfect rectangle, one that pleases the eye.
The golden ratio is found everywhere, it appears in nature as well as human made constructions. While it isn't as famous as pi, it is pretty important.
Most animals are built according to the golden ratio. Shells, seeds, pine cones, hurricanes and spiral galaxies are shaped according to the golden ratio. The number of flower petals on different flowers respond to the Fibonacci sequence. DNA follows the Fibonacci sequence! The golden ratio is everywhere! (15 examples)
And the Fibonacci sequence. The Fibonacci sequence is a sequence which Leonardo Fibonacci came up with when trying to calculate the ideal expansion pairs of rabbits over the course of one year. Each number is the sum of the two numbers that precede it, for example:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.
How is this related to the golden ratio?
Well, when you take a successive pair of numbers, the ratio is very close to the golden ratio. The larger the pair, the closer it is.
The golden ratio is thought of as the most pleasing to the eye. Many people try to use the golden ratio, such as the ancient Greeks with the Parthenon.
What I find is the most fascinating is the fact that the golden ratio is found everywhere in nature, everywhere you look. The golden ratio isn't close to a fraction like 3.14 is, it slips between fractions.
What do you admire about this golden ratio?

UPDATE: Next time, the Goldbach Conjecture and the Weak Goldbach Conjecture