Question

What’s the most ridiculous “proof left as an exercise” you’ve found?

Answer 1

Serge Lang left the Riemann Hypothesis as an exercise in his Complex Analysis (a later chapter on the Zeta function). He added, if you have troubles, ask your lecturer for help.

I also remember reading somewhere that famous mathematicians found a "Proof. Trivial." by Lang (his infamous Algebra textbook) was not trivial at all and published some papers out of it.

Answer 2

By ancient tradition, every textbook on computational complexity theory has an exercise equivalent to proving or disproving P=NP.

Answer 3

Stokes assigned the proof of Stokes' theorem on one of this exams, less than 5 years after he proved it himself.

Answer 4

I heard a story of a Professor lecturing who left a proof as an exercise but a student asked him to do it. He tried to oblige but couldn't remember how it was done. He apologized to the class and promised he'd come back with the proof next class. So he went down to the library and found a book that he knew would have the theorem, still mad at himself, he flipped to the correct chapter and found it...left as an exercise to the reader. And worse? He was the author of the book.

Answer 5

In my undergrad abstract algebra course, there was a lemma about prime ideals, and the proof was left as an exercise. After giving it some thought and realizing something doesn’t add up, I asked my professor, and he said that the lemma was just wrong and that was a mistake in the book.

Answer 6

There's an argument left to the reader in Milne's étale cohomology book. After getting stuck on it for a good long while, our professor emailed Milne himself to ask for a hint. He replied, "It's easy.... just think about it."

Answer 7

Katznelson’s Intro to Harmonic Analysis has some of the craziest exercises I’ve seen in my career. I have fond memories of our professor assigning problems that looked good, then one of them turned out to be a longstanding, historical, genuinely difficult problem. Decades in the making kind of problem. My proof for it was three cramped pages long, and only afterwards he said he wasn’t going to grade it for accuracy. Oof.

Answer 8

Maybe not necessarily "ridiculous" but there was this paper on arXiv which was one of our main sources for a seminar about geometric topology that contained numerous "left as an exercise" proofs which I found particularly annoying to be honest.

Answer 9

The Art of Computer Programming by Knuth has some great ones. Thankfully he was kind enough to rate their difficulty, so you knew what you were getting into!

Answer 10

There is the following unproven conjecture:

For every natural number *n* there exists another natural number *m* not equal to *n* such that *phi(n)=phi(m),*

where phi is eulers totient function.

This was an exercise in some number theory book because the author had a proof for it. It just turned out later that the proof suggested by the author was wrong.