These may be closer to the "every even positive integer is composite except 2" example, but for whatever they are worth: * Each positive integer n is the shortest side of an integer sided right triangle, except n = 1, 2, 4. * Every prime Fibonacci number has prime index, except 3, which has index 4=2*2. (here the Fibs start 1,1,2,3,5,8,13, ...) Jim -------------- On Tue, Oct 14, 2008 at 4:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Richard Guy wrote [as slightly reformatted]:
<< The corresponding problem where sums of {\em triples} of elements of a set are given has been settled by Boman & Linusson. The exceptions are precisely 3, 6, 27, 486.
For n=27 they give five examples of which the simplest is
{-4,-1^{10},2^{16}}
and its negative, where exponents denote repetitions.
For n=486 they give
{-7,-4^{56},-1^{231},2^{176},5^{22}}
and its negative.
This is so cool! I love results like this, where a simply stated problem has a straightforward answer, except for some small and/or strange set.
(It is surely no coincidence that these are 3, 2*3, 3^3, 2*3^5.)
I am interested to hear of mathematical results that have a small and strange set of exceptions.
A few I can think of are these:
* Any differentiable structure on n-space is equivalent to any other for all n except 4.
* The alternating group A_n is simple for all n except 4 (Hmmm, is there a connection here?)
* The automorphism group Aut(S^n) is isomorphic to S_n for all n except 2, 6.
* The ring of integers of the imaginary quadratic field Z(sqrt(-n)) has non-unique factorization except for n = 1, 2, 3, 7, 11, 19, 43, 67, 163.
* The ring of integers of the cyclotomic field Z(exp(2pi i/p) has non-unique factorization for all primes p except 2,3,5,7,11,13,19.
* There exists no real division algebra R^k for k = 2^n for all n except 0, 1, 2, 3.
Other such examples are solicited.
--Dan