The previous post showed how to find all groups whose order is a product of two primes using 2019 as an example. Here are a couple more observations along the same line, illustrating the odd Goldbach conjecture and Lagrange’s four-square theorem with 2019.

## Odd Goldbach Conjecture

**Goldbach’s conjecture** says that every even number greater than 2 can be written as the sum of two primes. The **odd Goldbach conjecture**, a.k.a. the **weak Goldbach conjecture**, says that every odd number greater than 5 can be written as the sum of three primes. The odd Goldbach conjecture isn’t really a conjecture anymore because Harald Helfgott proved it in 2013, though the original Goldbach conjecture remains unsettled.

The odd Goldbach conjecture says it should be possible to write 2019 as the sum of three primes. And in fact there are 2,259 ways to write 2019 as a non-decreasing sequence of primes.

3 + 5 + 2011 3 + 13 + 2003 3 + 17 + 1999 ... 659 + 659 + 701 659 + 677 + 701 673 + 673 + 673

## Lagrange’s four-square theorem

Lagrange’s four-square theorem says that every non-negative integer can be written as the sum of four squares. 2019 is a non-negative integer, so it can be written as the sum of four squares. In fact there are 66 ways to write 2019 as a sum of four squares.

0 1 13 43 0 5 25 37 0 7 11 43 ... 16 19 21 31 17 23 24 25 19 20 23 27

Sometimes there is a unique way to write a number as the sum of four squares. The last time a year could be written uniquely as a sum of four squares was 1536, and the next time will be in 2048.