Re: [math-fun] what's the best way of finding n unknown integers from their sums taken k at a time
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 _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
On Tue, Oct 14, 2008 at 2:29 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Other such examples are solicited.
All evens are composite except 2 ;) -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com
On Tuesday 14 October 2008, Dan Asimov wrote:
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. ... 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: ... Other such examples are solicited.
Classification of finite simple groups. (A few nice "obvious" families, plus the sporadic ones.) I'm not convinced by your examples of the form "Such-and-such a ring always has non-unique factorization, except for these cases"; this seems a bit like Mike Stay's joke example "all even numbers are composite, except 2". I mean, surely the real story for k(sqrt(d)), for instance, is that the class number function -> oo as d -> oo, so obviously it's only 1 for a finite number of cases. -- g
I like the Johnson solids. 92 solids + prisms + antiprisms. ----- Original Message ----- From: "Gareth McCaughan" <gareth.mccaughan@pobox.com> To: <math-fun@mailman.xmission.com> Sent: Tuesday, October 14, 2008 6:34 PM Subject: Re: [math-fun] what's the best way of finding n unknown integersfrom their sums taken k at a time
On Tuesday 14 October 2008, Dan Asimov wrote:
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. ... 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: ... Other such examples are solicited.
Classification of finite simple groups. (A few nice "obvious" families, plus the sporadic ones.)
I'm not convinced by your examples of the form "Such-and-such a ring always has non-unique factorization, except for these cases"; this seems a bit like Mike Stay's joke example "all even numbers are composite, except 2". I mean, surely the real story for k(sqrt(d)), for instance, is that the class number function -> oo as d -> oo, so obviously it's only 1 for a finite number of cases.
-- g
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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
participants (5)
-
Dan Asimov -
David Wilson -
Gareth McCaughan -
James Buddenhagen -
Mike Stay