# Partition function 2

2. In additive number theory the basic problem is how an integer can be written as the sum of different integers.

And as an example we can take one of the most famous unsolved problems in mathematics,

Goldbach conjecture: Every even integer $n>2$ is the sum of two odd primes.

Other terms related to the conjecture are,

Goldbach number: All those positive integers that can be expressed as the sum of two odd primes. And the smallest number that can be written in such a way is $4$. So, every even $n>4$ are Goldbach numbers.

Goldbach Partitions: The expression is called the Goldbach partition of that number.

1. The partition function I wrote about last month is just one of such problems in additive number theory, which tells us the number of ways an integer can be written as the sum of different positive integers.

$P(1) = 1$

$P(2) = 2 = 1+1$

$P(3) = 3 = 1+2 = 1+1+1$

This particular problem has a specific name, it’s called unrestricted partitions.

Because the number of integers that can be used to write the sum is unrestricted. Repetition is allowed and the order of numbers does not matter.

So, $P(n)$ is called unrestricted partition function.