Results that seem trivial, but are not

Question

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?

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 09 '18 at 16:22
Possible duplicate of "It looks straightforward, but actually it isn't", or What seemingly innocuous results in mathematics require advanced proofs? — Arnaud D., Apr 10 '18 at 09:21

score 3 · Answer 1 · answered Apr 09 '18 at 16:16

3

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.

answered Apr 09 '18 at 16:16

J.G.

115,835

Dietrich Burde · Answer 2 · 2018-04-09T18:04:25.703

2

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.

edited Apr 09 '18 at 18:04

answered Apr 09 '18 at 16:30

Dietrich Burde

130,978

2

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 09 '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 09 '18 at 18:05

Results that seem trivial, but are not

2 Answers2