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awarded  Yearling
Aug
7
comment When is $(q^k-1)/(q-1)$ a perfect square?
I seemed to recall it being proved in Ribenboim's book about Catalan's conjecture (published before Mihailescu solved it), and, indeed, Shorey writes that it is (on p. 111, apparently) in his MathSciNet review of the book: ams.org/mathscinet-getitem?mr=95a:11029.
Jun
30
comment Linear transformation of a polyhedron
Indeed, 'dualization' (polar reciprocation) provides a geometric proof of the equivalence, albeit a somewhat involved one. Off the top of my head, I think Stoer and Witzgall present that approach in their book on "Convexity and optimization" but I do not have it handy to check at the moment.
Jun
29
comment Linear transformation of a polyhedron
What's wrong with the following? (Probably very close to what @Misha had in mind.) View a (not necessarily bounded) polytope as $\text{Conv}(\{u_i\}_{1 \le i \le k})+\text{Cone}(\{v_i\}_{1 \le i \le m})$ where $\text{Cone}(\{v_i\}_{1 \le i \le m}) = \left\{ \sum_{i=1}^m t_i v_i, \, t_i \ge 0 \right\}$. Now, if $L \in \text{End}(\mathbb{R}^n)$, then $L(P) = \text{Conv}(\{L(u_i)\}_{1 \le i \le k})+\text{Cone}(\{L(v_i)\}_{1 \le i \le m})$, which yields (1) (and so (2)).
Oct
13
awarded  Caucus
Sep
6
comment Generalized quasi-perfect numbers
I wondered about this a while ago and dblues had some nice preliminary observations in the very last post of this AoPS thread: artofproblemsolving.com/Forum/viewtopic.php?f=57&t=83696
Jul
17
awarded  Commentator
Jul
17
comment Integer-distance sets
@Joseph O'Rourke: The answer is the same in $\mathbb{R}^d$ with $d \ge 3$. See e.g. the original paper by N. H. Anning and P. ErdÅ‘s: ams.org/journals/bull/1945-51-08/S0002-9904-1945-08407-9/…
Jul
17
comment Integer-distance sets
@Carl: It is not an unsolved problem anymore. Such a set has been constructed by T. Kreisel and S. Kurz in 2008: arxiv.org/abs/0804.1303v1
Jul
16
comment Generalization of Rado's Single Equation theorem
See also this nicely presented expository note by T. Tao (p. 12 in particular): math.ucla.edu/~tao/preprints/Expository/ramsey.dvi
Jun
6
awarded  Nice Answer
Apr
2
comment sum of digits in different bases
For 1000: our maximal $n$ must satisfy $(s_10(n)=) s_2(n) \le 9$. There are only 5 numbers with 9 ones in binary --- none of them works. Then check those numbers whose decimal sum is 8 (800, 710, etc.). None of them work until 503, which is the answer in this case. Granted, this is still somewhat brute force...
Mar
22
answered Rational points on a sphere in $\mathbb{R}^d$
Jun
16
awarded  Enthusiast
Jun
7
comment determinants and polynomials in matrices
For discussions of it, I highly recommend the following three threads on the AoPS forum: artofproblemsolving.com/Forum/viewtopic.php?f=349&t=43292 artofproblemsolving.com/Forum/viewtopic.php?f=349&t=218429 artofproblemsolving.com/Forum/viewtopic.php?f=349&t=279631
Jun
4
comment Formula in common: How to search for same/similar equations in other knowledge domains?
There was a long discussion touching on that issue over at Tim Gowers's blog: gowers.wordpress.com/2012/03/21/…
May
29
comment Bounding a signed sum of complex numbers
There is no need to (re)post it on the AoPS forum: artofproblemsolving.com/Forum/viewtopic.php?f=42&t=386586
May
20
comment Estimate for products of integers that are relatively prime with $N$
Yes, mod N indeed, thank you.
May
20
comment Elementary applications of Krein-Milman
A reference for that proof is the book "A Problem Seminar" by Donald J. Newman.
May
20
answered Estimate for products of integers that are relatively prime with $N$