TITLE: What can be proven in Peano arithmetic but not Heyting arithmetic?
QUESTION [25 upvotes]: Hi. I'll confess from the start to not being a logician. In fact this question came up not from research but during a discussion with a friend about whether the classical proof that $\sqrt{2}$ is irrational can be made acceptable to an intuitionist. (It can be.)
The question is: Are there any "natural" statements which can be proven in Peano Arithmetic, but not in Heyting Arithmetic (Peano Arithmetic but with a logic that does not admit the law of the excluded middle)? 
In fact, any statements -- even pathological ones -- that can be proven in one but not the other would be interesting to me, since I wasn't able to come up with any. (Even after doing a few web searches!) But of course, the closer to the surface the better.

REPLY [15 votes]: According to Harvey Friedman, the following theorem is provable in PA but not HA:

Every polynomial $P:\mathbb{Z}^n \to \mathbb{Z}^m$ with integer coefficients assumes a value closest to the origin.

That is, there is a value which is at least as close to the origin, in the
Euclidean distance, than any other value.  This is unprovable in HA even when $m=1$.