\[\Large (1-x)(1-x^2)(1-x^3) \cdots = \sum_{n=-\infty}^\infty (-1)^n x^{n(3n+1)/2} \]

Anyone have a proof of this?

\[\Large (1-x)(1-x^2)(1-x^3) \cdots = \sum_{n=-\infty}^\infty (-1)^n x^{n(3n+1)/2} \]

Anyone have a proof of this?

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TopNewestThis 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. – Michael Mendrin · 8 months, 1 week ago

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ive read it in \(an\quad introduction\quad to\quad the\quad theory\quad of\quad numbers\quad\) by hardy and i was lost (he`s awesome) – Hummus A · 8 months, 1 week ago

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