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Goldbach's Conjecture

We have seen that a prime number is one whose only factors are itself and 1. Mathematicians often exclude 1 itself from the list of prime numbers because 1 is an exception for many theorems (mathematical results that have been proved) about prime numbers.
The prime numbers up to 100 are:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
One fact that has been proved is that Every Number is the Product of Prime Numbers is just one way.
By just one way we mean that if we list the prime factors in order, allowing repeated numbers, then there is only one unique collection of such prime numbers for every single whole number. For instance, 12 = 2x2x3 as a product of prime numbers, and no other set of prime numbers will have 12 as its product.
What about adding prime numbers rather than multiplying them?

Christian Goldbach (1690-1764), a Professor of Mathematics at the Russian Imperial Academy, thought about this question.
The simplest case is to use just two prime numbers in a sum. So for instance here are some small numbers written as the sum of two primes:
4=2+2, 5=2+3, 6=3+3, 7=2+5, 9=2+7, 12=7+5

However, we cannot find two primes with a sum of 27.
Because all primes except 2 are odd, adding two primes none of which is 2, is odd+odd which must be even. However, all primes except two are odd - which makes 2 the oddest!! So Goldbach concentrated just on even numbers and asked the question:
Is every even number the sum of two odd primes?
He thought the answer was always "Yes" but could not prove it. The result is therefore called:

Goldbach's Conjecture
Every even number bigger than 6 is the sum of two odd primes.

Here are some of the smaller even numbers written as the sum of two odd primes:
 
 4 = 2+2 -- oops!  these are both even!
 6 = 3+3
 8 = 3+5
10 = 3+7 = 5+5
12 = 5+7
14 = 3+11 = 7+7
16 = 3+13 = 5+11
18 = 5+13 = 7+11
20 = 3+17 = 7+13
22 = 3+19 = 5+17 = 11+11
24 = 5+19 = 7+17 = 11+13
26 = 3+23 = 7+19 = 13+13
28 = 5+23 = 11+17
30 = 7+23 = 11+19 = 13+17
Can you...
  1. ...continue this list of even numbers up to 50?
  2. ...find any even number below 50 that is not the sum of two odd primes?
  3. ...find the number which has the biggest collection of pairs of odd primes that sum to it?
  4. ...find another ODD number which is not the sum of two prime numbers (so 2 would have to be one of your prime numbers)?
  5. ...continue your list up to 100 and answer all these questions again for the extended list?


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Primes, Factors and Divisibility