Question

Write a (short) report on the uses of writing a positive integer as a sum of four squares.

One paragraph or two paragraphs.

Don't copy from Google, use your own words.

Type it on word document.

Answer 1

The problem: we want to write a positive integer as a sum of four squares (of integers). Why we want to do this? Because its an interesting and challenging problem. Some notable small examples are and

We have found many integers which can be written as sum of four squares. So the natural question is, which are more? Can all integers be written as sum of four squares. As the numbers get larger, the "blocks" of numbers which must be squares also diversify in size. So we expect it to be the case that numbers can be written as sum of four squares. Lagrange was the first person to prove this in 1770. His proof runs into several pages and uses some advanced ideas. There are also many modern and sophisticated algorithms to find the representation of any natural number n as a sum of four squares.