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Goldbach's conjecture is one of the oldest unsolved problems in mathematics. It states:

   Every even integer greater than 2 can be expressed as the sum of two primes.

(Remember that a prime is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first prime numbers are 2, 3, 5, 7, 11, 13, 17, ...)

For the first even numbers, the Goldbach's conjecture reads:

  • 4 = 2 + 2
  • 6 = 3 + 3
  • 8 = 3 + 5
  • 10 = 3 + 7
  • 12 = 5 + 7
  • 14 = 7 + 7
  • 16 = 5 + 11

Note that for some numbers there are more than one way to express them. For instance, the number 10 can be written as 3+7, but also as 5+5.

Mathematicians believe that the Goldbach's conjecture is always true for any even number greater than 2. Christian Goldbach and Leonhard Euler thought the same thing in 1742, when they proposed the conjecture. Euler wrote:

   I regard it as a completely certain theorem, although I cannot prove it.

Nowadays, it remains unproved. Modern computers have checked it for every even number up to 4,000,000,000,000,000,000 but that is not enough, as there are infinitely many numbers. So, there are two options to close the problem:

  1. Find a large even number that cannot be expressed as the sum of only two primes. (That's unlikely. Mathematicians believe there is no such a number)
  2. A sophisticated mathematical reasoning to ensure that the conjecture is true for EVERY even number greater than two.

 

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References:

http://en.wikipedia.org/wiki/Goldbach%27s_conjecture

http://www.segerman.org/autologlyphs/names-numbers.jpg

 

 

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