How to prove Fermat's Triangular Number Theorem/Gauss's Eureka Theorem?

[ArenasField]

The theorem states that every positive integer is either a triangular number, the sum of two triangular numbers, or the sum of three triangular numbers, where triangular numbers are given by T=(x^+x)/2.

My question is, does anyone have a proof or can anyone show me the proof for this? Even if it isn't a complete proof, could someone please share it with me? I can't find it anywhere on the internet, but I know that Gauss proved it a few centuries ago.

I've been looking at the first 100 integers, and I've proven it for all of them. I've also determined how to find whether or not an integer is a triangular number, but I haven't been able to prove the theorem.

Can someone please help me? Sorry if this question isn't in the right forum, or if it's not a very good/concise question.

Thanks in advance

 

[Romsek]

The theorem states that every positive integer is either a triangular number, the sum of two triangular numbers, or the sum of three triangular numbers, where triangular numbers are given by T=(x^+x)/2.

My question is, does anyone have a proof or can anyone show me the proof for this? Even if it isn't a complete proof, could someone please share it with me? I can't find it anywhere on the internet, but I know that Gauss proved it a few centuries ago.

I've been looking at the first 100 integers, and I've proven it for all of them. I've also determined how to find whether or not an integer is a triangular number, but I haven't been able to prove the theorem.

Can someone please help me? Sorry if this question isn't in the right forum, or if it's not a very good/concise question.

Thanks in advance

here's a proof of the general polygonal number theorem, you could have googled this yourself.

http://www.ams.org/journals/proc/1987-099-01/S0002-9939-1987-0866422-3/S0002-9939-1987-0866422-3.pdf

 

[ArenasField]

here's a proof of the general polygonal number theorem, you could have googled this yourself.

http://www.ams.org/journals/proc/1987-099-01/S0002-9939-1987-0866422-3/S0002-9939-1987-0866422-3.pdf

I have already found that link, but it just states "Gauss [4] proved that every nonnegative integer is the sum of three triangular number." And I don't quite see how the theorems shown apply to my question. Any help please?

Is there a simpler way to prove my question?