SOLUTION: One of the most famous unsolved problems in mathematics is a conjecture
made by Christian Goldbach in a letter to Leonhard Euler in 1742. The
conjecture made in this letter is no
Algebra.Com
Question 500690: One of the most famous unsolved problems in mathematics is a conjecture
made by Christian Goldbach in a letter to Leonhard Euler in 1742. The
conjecture made in this letter is now known as Goldbach’s Conjecture. The
conjecture is as follows:
Every even integer greater than 2 can be expressed as the sum of two
(not necessarily distinct) prime numbers.
Currently, it is not known if this conjecture is true or false, although most
mathematicians believe it to be true.
(a) Describe one way to prove that Goldbach’s Conjecture is false.
(b) Prove the following:
If there exists an odd integer greater than 5 that is not the sum of
three prime numbers, then Goldbach’s Conjecture is false.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
1. Find an even integer greater than 2 that cannot be expressed as the sum of two prime numbers.
2. Let 
be an odd integer greater than 5 such that three not necessarily distinct prime numbers that sum to do not exist. Let 
be a prime number greater than 2, then must be odd. The difference of any two odd integers is even \ -\ \left(2n\ -\ 1\right)\ =\ (2m\ -\ 2n)\right))
. Hence, 
is even. But Goldberg's conjecture says that must be the sum of two primes, and therefore must be the sum of three primes. Therefore, if exists, there must be a that is even and not the sum of two primes proving Goldberg's Conjecture false.
John
My calculator said it, I believe it, that settles it
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