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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.
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.
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
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This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.
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2^{34}
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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.
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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.
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.
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
Every integer is the sum of 3 triangular numbers, proven by Gauss, 7/10/1796. Ed Gray