Struggling to prove 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?

#Algebra

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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.

Michael Mendrin - 5 years, 4 months ago

thanks!

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)

Hamza A - 5 years, 4 months ago

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.

Michael Mendrin - 5 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$