In 1742 Christian Goldbach sent the above letter to Euler posing the problem:

**Goldbach conjecture:** *Every even integer greater than 4 can be written as the sum of two odd primes.*

Let’s try this out

6 = 3 + 3

8 = 3 + 5

10 = 3 + 7

12 = 5 + 7

14 = 7 + 7

16 = 3 + 13

Keep on going with this and you will see that writing a large even number as the sum of two odd primes gets harder and harder to do. Does it always work? This is an open question called the *Goldbach Conjecture*. Certainly computers have verified that the Goldbach conjecture is true for even numbers less than 1.6 x 10^(18) – definitely a pretty big number. But there is always the possibility that there is some really really BIG even number out there which can *not* be written as the sum of two odd primes.

Here are some interesting attempts at this problem:

(1938) Esterman proved that E(x)/x goes to zero as x goes to infinity. Here E(x) is the number of even numbers not exceeding x that can NOT be written as the sum of two primes. This result shows that “most” (in some probabilistic sense) even numbers can be written as the sum of two primes.

(1933) Shnirelman proved than every number (not just even) can be written as the sum of at most 20 primes.

(1995) Ramare proved that every even number is the sum of at most 6 primes.

(1995) Kaniecki proved that assuming the Generalized Riemann Hypothesis then every odd number (greater than one) is the sum of at most 5 primes.

(2012) Terrance Tao removed the dependence on the GRH and showed that every odd number (greater than one) is the sum of at most 5 primes.

There is a weaker version of the Goldbach conjecture where mathematicians have made some progress.

**Weak Goldbach conjecture:** *Every odd number greater than 7 can be written as the sum of three odd primes.*

Let’s try this one out.

9 = 3 + 3 + 3

11 = 3 + 3 + 5

13 = 3 + 5 + 5

15 = 5 + 5 + 5

17 = 5 + 5 + 7

19 = 5 + 7 + 7

21 = 7 + 7 + 7

As you can imagine, this gets harder and harder to do for really BIG odd numbers. Can we always do this?

**Theorem:** *If the Goldbach conjecture is true than then the weak Goldbach conjecture is true.*

**Proof:** If every even number greater than 4 is the sum of two odd primes (the Goldbach conjecture), merely adding three to each even number greater than 4 will produce the odd numbers greater than 7 (the weak Goldbach conjecture). **QED.**

Here is what is amazing about the weak Goldbach conjecture. You would think that a computer could verify the result for “small” odd numbers and then some sort of theory would take over for all odd numbers. The results are the opposite. Here is my favorite theorem in all of mathematics.

**Theorem (Vinogradov 1937)**:* Any odd number larger than 10^(7,000,000) is the sum of three odd primes.*

Let’s get a handle on the size of that number. The number of particles in the universe is about 10^(80). Ten to the seven million power is just an amazing number! So, you think, so the result is true for all large odd numbers, we just need to break out our laptops and start working on the “small” odd numbers. How hard can this be? Well, if 10^(80) is the number of particles in the universe, then certainly no physical computer could so this computation since 10^(7,000,000) is more particles than you have to work with. There is some “good” news here. Chen was able to show that Vinogradov’s insane number could be reduced to 10^(7194) (i.e., every odd number greater than 10^(7194) can be written as the sum of three odd primes). However, even this breathtaking improvement is still not enough to put the problem into computing range.

(1997) Deshouillers, Effinger, te Riele and Zinoviev showed that assuming the GRH that weak Goldbach is true.