Nui nā hōʻoia o ka pau ʻole o nā helu mua-ʻaʻole ʻike ʻia ka theorem Euclid kaulana loa mai ka Book of Elements i kekahi papa kumumanaʻo helu maʻamau. I ka American Matematika Makahiki (hoʻopuka 122) i ka makahiki 2015 ua paʻi ʻo Sam Northshield i kahi hōʻoia hoʻoleʻaleʻa haʻahaʻa ma ke ʻano o ka laina hoʻokahi, kahi aʻu e makemake ʻole ai e kāohi iā ʻoe (me nā ʻōlelo pōkole).
$$0 < \prod_{p} \sin \underbrace{ \left( \frac{\pi}{p} \right) }_{ < \pi, \text{ da } p > 1 } = \prod_{p} \sin \left( \frac{\pi}{p} + 2 \pi \underbrace{ \frac{ \prod_{p'} p' }{p} }_{ \in \mathbb{N} } \right) = \prod_{p} \sin \left( \pi \underbrace{ \frac{ 1 + 2 \prod_{p'} p' }{p} }_{ \in \mathbb{N}, \text{ da } \left( 1 + 2 \prod_{p'} p' \right) \notin \mathbb{P} } \right) = 0$$