# Sum of Triangular Numbers

Is the set of all positive integers which cannot be expressed as a sum of distinct triangular numbers finite?

Note by Marta Reece
1 year, 4 months ago

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I think that after the number $$33$$, all positive integers can be expressed as a sum of $$3$$ or less distinct triangular numbers. $$2, 5, 8, 12, 23, 33$$ can not be expressed as a sum of distinct triangular numbers. $$20$$ can be expressed as a sum of $$4$$ distinct triangular numbers.

- 2 weeks, 3 days ago

Check this out

Fermat Polygonal Number Theorem

But of course, the numbers aren't necessarily distinct, so that's an extra wrinkle. At least this reduces the problem into determining whether number of the form 2T, where T is a triangular number, can be represented by 2 or more other triangular numbers.

- 1 year, 3 months ago

This definitely cannot be done for some numbers. For example 5 = 3 + 1 + 1, but there is no expression for it in terms of unique triangular numbers.

- 1 year, 3 months ago

Every integer is the sum of three triangular numbers, proven by Gauss, and entered in his notebook dated 7/ 10/ 1796. It is sometimes called the Eureka Theorem, for that's what Gauss called it. Ed Gray

- 8 months ago

Every integer is the sum of 3 triangular numbers, proven by Gauss, 7/10/1796. Ed Gray

- 8 months ago