MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am interested in theorems with unexpected conclusions. I don't mean an unintuitive result (like the existence of a space-filling curve), but rather a result whose conclusion seems disconnected from the hypotheses. My favorite is the following. Let $f(n)$ be the number of ways to write the nonnegative integer $n$ as a sum of powers of 2, if no power of 2 can be used more than twice. For instance, $f(6)=3$ since we can write 6 as $4+2=4+1+1=2+2+1+1$. We have $(f(0),f(1),\dots) = $ $(1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,\dots)$. The conclusion is that the numbers $f(n)/f(n+1)$ run through all the reduced positive rational numbers exactly once each. See A002487 in the On-Line Encyclopedia of Integer Sequences for more information. What are other nice examples of "unexpected conclusions"?

share|cite|improve this question

closed as no longer relevant by Felipe Voloch, Bill Johnson, Andrés E. Caicedo, Mark Sapir, Ryan Budney Jan 5 '12 at 21:50

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

15  
Very interesting question! But since it has no right answer, and you are asking for a big list, I think it should be community wiki. – Grétar Amazeen Mar 13 '10 at 21:32
4  
I'll send you a shirt: thenerdiestshirts.com/blog/math-shirt-cw/#more-116 – Douglas Zare Mar 13 '10 at 21:48
2  
Thanks for great shirt! I guess it's difficult to make precise the difference between "unintuitive" and "unexpected conclusion." Some examples are at math.dartmouth.edu/~pw/solutions.pdf. Another nice example is the cake icing problem (demonstrations.wolfram.com/TheCakeIcingPuzzle). – Richard Stanley Mar 13 '10 at 22:47
11  
Stolen from Kevin Buzzard's comment at mathoverflow.net/questions/15050/linear-algebra-problems: If A and B are real n x n matrices, A^2+B^2=AB, and AB-BA is invertible, then n is a multiple of 3. – Jonas Meyer Mar 13 '10 at 23:14

33 Answers 33

Given a $C^{\infty}$ function $f(x)$, let $\Delta_n$ be the difference between the integral of $f(x)$ and its $n$'th Riemann sum: $$\Delta_n = \int_0^1 f ~dx ~-~ \sum_{i=1}^{n} f(i/n) \frac{1}{n}$$ Clearly, $\Delta_n$ goes to $0$, but at what rate? Its easy to see that its possible for $\Delta_n$ to decay as $\Theta(1/n)$, e.g consider what happens with $f(x)=x$.

Theorem: If $f(x)$ is periodic with period $1$, then $\Delta_n$ decays faster than any polynomial in $n$.

share|cite|improve this answer
    
It looks surprisingly! But estimations from the cited paper includes the uniform norm of $n$-th derivative, so I could not see... – Petya Mar 14 '10 at 22:32
    
Right - upper bounds on $\Delta_n$ have to depend on the function $f$ somehow. Here is a restatement of my comment in messier way that, however, makes this dependence explicit, and hopefully answers your question. For an arbitrary $C^{\infty}$ function $f$, one can prove an upper bound of $\Delta_n \leq C(f)/n$, where $C(f)$ is some functional. In fact, its obvious that $C(f) = \max_{x \in [0,1]} f'(x)$ works. The example of $f(x)=x$ shows that if $k>1$ then one cannot prove a bound of the form $\Delta_n \leq C_k(f)/n^k$. – alex Mar 15 '10 at 0:02
    
...by contrast, if $f(0)=f(1)$, then a bound of the form $\Delta_n \leq C_k(f)/n^k$ can be proven for any $k>1$. Thus in this case $\Delta_n$ decays faster than $1/n^k$ for any $k$. – alex Mar 15 '10 at 0:05
2  
You need more than that $f(0)=f(1)$. For $x(1-x)$ the error is quadratic. Perhaps you want $f$ to be periodic with period $1$, and still smooth at $0$. – Douglas Zare Mar 15 '10 at 0:57
    
...er, yes. Serves me right for carelessly replacing the condition "$f$ is periodic" with "$f(0)=f(1)$." – alex Mar 15 '10 at 1:15

The classical differential geometry results should definitely be mentioned here. Although it may seem not surprising for us, Gauss found his Theorema Egregium to be truly remarkable and unexpected. My favorite example is Gauss-Bonnet Theorem.

share|cite|improve this answer

Not the answer you're looking for? Browse other questions tagged or ask your own question.