Hacker News with Generative AI: Proofs

Anatomy of a Formal Proof (ams.org)
It has been a long day and you are making your way through a paper related to your work. You suddenly come across the following remark: “…since $x$ and $y$ are eigenvectors of $f$ with distinct eigenvalues, they are linearly independent.” Wait—how does the proof go? You should really know this. Here $x$ and $y$ are nonzero elements of a vector space $V$ and $f : V to V$ is a linear map.
Where do those undergraduate divisibility problems come from? (grossack.site)
Oftentimes in your “intro to proofs” class or your first “discrete math” class or something similar, you’ll be shown problems of the form “prove that for $n^6 + n^3 + 2n^2 + 2n$ is a multiple of $6$ for every $n$”… But where do these problems come from? And have you ever stopped to think how magical this is?
Machine-Assisted Proof [pdf] (ams.org)
A visual proof that a^2 – b^2 = (a + b)(a – b) (futilitycloset.com)
A visual proof that a2 – b2 = (a + b)(a – b).
Grothendieck’s use of equality (arxiv.org)