# Results that seem trivial, but are not [duplicate]

I saw the Cayley-Hamilton theorem

Let $\mathbf{A}$ be a $n\times n$-matrix, and $p(\lambda)=\det(\lambda \mathbf{I}_n-\mathbf{A})$ the characteristic polynomial of $\mathbf{A}$. Then $p(\mathbf{A})=\mathbf{0}$

and I thought it was trivial since $\det(\mathbf{A}-\mathbf{A})=0$. I checked with Wikipedia and learned that this didn't work because $\lambda$ is a scalar, and $\det(\mathbf{0})$ is also a scalar, and not a matrix. This brings me to my question:

Are there any other results that seem trivial, but aren't?

## marked as duplicate by user334732, Arnaud D., Community♦Apr 10 '18 at 9:52

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• Queen Dido's approach to the isoperimetric problem seems relevant. Richard Tapia wrote (2013), "When first introduced to this problem, even the less mathematically initiated individuals readily conjecture that the solution is the circle." However the history of attempts at a fully rigorous proof cover two thousand years. – hardmath Apr 9 '18 at 16:22
• – Arnaud D. Apr 10 '18 at 9:21

## 2 Answers

The Jordan curve theorem states that, if you draw a closed non-intersecting curve in the plane, it divides the plane into those points that are inside the curve, outside it or on it.

The weak Goldbach conjecture (a theorem now, it seems, proved by Harald Helfgott): Every odd integer $n>5$ is a sum of three primes. If you try yourself examples, this seems trivially true. There are millions of possibilities for a big odd $n$. Nevertheless, the proof is extremely complicated.

• In case anyone doesn't know, it was proved in 2013 by Harald Helfgott. (I looked it up because I didn't realise it had even been proven.) – J.G. Apr 9 '18 at 16:47
• Yes, there are $5$ arXiv papers, see wikipedia. However, an article with a proof has not been published in a journal yet. – Dietrich Burde Apr 9 '18 at 18:05