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2 \times 3
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This is the famous Euler's Pentagonal Number Theorem, and while there are many published proofs of it, none of them are short and easy. However, this one is not bad.
As might be expected, this is related to the partition function, since we are looking at how many of the products of powers of x cancel each other out.
Euler took something like 10 years to solve this one, so don't feel bad if you can't figure it out right away or anytime soon.
I see it now on page 264 of Hardy's book. It does contain an "elementary proof due to Franklin" which runs about a page or so explaining how those terms cancel out, except for the pentagonal number powers.
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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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to ensure proper formatting.2 \times 3
2^{34}
a_{i-1}
\frac{2}{3}
\sqrt{2}
\sum_{i=1}^3
\sin \theta
\boxed{123}
Comments
This is the famous Euler's Pentagonal Number Theorem, and while there are many published proofs of it, none of them are short and easy. However, this one is not bad.
Pentagonal Number Theorem
As might be expected, this is related to the partition function, since we are looking at how many of the products of powers of x cancel each other out.
Euler took something like 10 years to solve this one, so don't feel bad if you can't figure it out right away or anytime soon.
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thanks!
ive read it in anintroductiontothetheoryofnumbers by hardy and i was lost (he`s awesome)
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I see it now on page 264 of Hardy's book. It does contain an "elementary proof due to Franklin" which runs about a page or so explaining how those terms cancel out, except for the pentagonal number powers.