[math-fun] Zhang's proof of the bounded prime gap theorem
Emanuel Kowalski's blog gives a useful summary: http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/
http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-t... On 5/23/13, Victor Miller <victorsmiller@gmail.com> wrote:
Emanuel Kowalski's blog gives a useful summary:
http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
http://www.newscientist.com/article/dn23644-game-of-proofs-boosts-prime-pair... On 6/4/13, Fred lunnon <fred.lunnon@gmail.com> wrote:
http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-t...
On 5/23/13, Victor Miller <victorsmiller@gmail.com> wrote:
Emanuel Kowalski's blog gives a useful summary:
http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Polyomino non-tiling problem: Definition: A p-dimensional n-omino X is a union of integer lattice cubes (of size 1^p) such that i) The interior of X is connected, AND ii) X is topologically equivalent to a closed p-dimensional disk. [Note: I'm adding the nonstandard condition ii) in order to avoid trivial cases of non-tiling due to holes, such as the heptomino with its squares' centers arranged thus: * * * * * * * .] In p-dimensional Euclidean space, find the least N for which there exists a p-dimensional N-omino, copies of which cannot tile p-space. (E.g., it's known that F(2) = 7.) Call this N by the notation F(p) QUESTION I): Can F(p) be determined explicitly for all p ??? ------------- QUESTION II): In any case, can an asymptotic formulas be found ------------- for F(p), as p -> oo ??? --Dan
It's easy to show, for p>2, F(p)<= 2 3^(p-1) - 1. -Veit On Jun 13, 2013, at 2:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Polyomino non-tiling problem:
Definition: A p-dimensional n-omino X is a union of integer lattice cubes (of size 1^p) such that
i) The interior of X is connected,
AND
ii) X is topologically equivalent to a closed p-dimensional disk.
[Note: I'm adding the nonstandard condition ii) in order to avoid trivial cases of non-tiling due to holes, such as the heptomino with its squares' centers arranged thus:
* *
* *
* * *
.]
In p-dimensional Euclidean space, find the least N for which there exists a p-dimensional N-omino, copies of which cannot tile p-space.
(E.g., it's known that F(2) = 7.)
Call this N by the notation F(p)
QUESTION I): Can F(p) be determined explicitly for all p ??? -------------
QUESTION II): In any case, can an asymptotic formulas be found ------------- for F(p), as p -> oo ???
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
You should add "allowing tiles to be turned over, reflected, rotated, etc." to the rules. If not, the 2D answer changes. Another likely 3D issue is whether you require that the tiling be physically assembleable, without requiring some temporary interpenetration. Rich ----- Quoting Veit Elser <ve10@cornell.edu>:
It's easy to show, for p>2, F(p)<= 2 3^(p-1) - 1.
-Veit
On Jun 13, 2013, at 2:54 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Polyomino non-tiling problem:
Definition: A p-dimensional n-omino X is a union of integer lattice cubes (of size 1^p) such that
i) The interior of X is connected,
AND
ii) X is topologically equivalent to a closed p-dimensional disk.
[Note: I'm adding the nonstandard condition ii) in order to avoid trivial cases of non-tiling due to holes, such as the heptomino with its squares' centers arranged thus:
* *
* *
* * *
.]
In p-dimensional Euclidean space, find the least N for which there exists a p-dimensional N-omino, copies of which cannot tile p-space.
(E.g., it's known that F(2) = 7.)
Call this N by the notation F(p)
QUESTION I): Can F(p) be determined explicitly for all p ??? -------------
QUESTION II): In any case, can an asymptotic formulas be found ------------- for F(p), as p -> oo ???
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
In the following, "tile" means p-dimensional n-omino. Yes, I did mean that any tile isometric to another tile is to be considered the same. (This includes mirror images.) There is no requirement about physical assemblability. As usual, it is assumed that in any tiling the interiors of any two tiles are disjoint. Simpler than the phrasing of the definition below, but equivalent: We assume each tile (p-dimensional n-omino) in any position is the union T of n closed unit lattice cubes such that T is topologically equivalent to a p-dimensional closed disk. (I.e., topologically equivalent to the closed unit p-cube [0,1]^p.) This definition excludes conventional multiply-connected polyominoes, which are no doubt interesting to ask similar questions about, but it keeps the question at hand simpler. (For instance, consider the 3D octomino Q consisting of a 3x3x1 arrangement of unit cubes, except for the middle one. The only way to fill the hole in Q is with another, linked, copy of Q. (Call such a union of two linked copies of Q by the name "Q2". (Then: Can such 16-ominoes Q2 tile 3-space? Each of these is topologically equivalent to a closed unit p-cube, so fits the question at hand.) --Dan ------------------------------------------------------------------ Rich writes: << You should add "allowing tiles to be turned over, reflected, rotated, etc." to the rules. If not, the 2D answer changes. Another likely 3D issue is whether you require that the tiling be physically assembleable, without requiring some temporary interpenetration. . . . I wrote: << Polyomino non-tiling problem: Definition: A p-dimensional n-omino X is a union of integer lattice cubes (of size 1^p) such that i) The interior of X is connected, AND ii) X is topologically equivalent to a closed p-dimensional disk. . . . In p-dimensional Euclidean space, find the least N for which there exists a p-dimensional N-omino, copies of which cannot tile p-space. (E.g., it's known that F(2) = 7.) Call this N by the notation F(p) QUESTION I): Can F(p) be determined explicitly for all p ??? ------------- QUESTION II): In any case, can an asymptotic formulas be found ------------- for F(p), as p -> oo ???
From Wikipedia: ----- Saturday Review reached its maximum circulation of 660,000 in 1971. Ironically, its decline began in the same year. ----- --Dan
here's another link to more recent work: http://terrytao.wordpress.com/2013/06/04/online-reading-seminar-for-zhangs-b... zhang's 70,000,000 has now been reduced to 4,801,744 (as of 8:15 pm eastern time!). in the first paragraph of the above webpage, tao gives a link to a this, http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_pri... which has the current record low number. bob baillie ----- Fred lunnon wrote:
http://terrytao.wordpress.com/2013/06/03/the-prime-tuples-conjecture-sieve-t...
On 5/23/13, Victor Miller <victorsmiller@gmail.com> wrote:
Emanuel Kowalski's blog gives a useful summary:
http://blogs.ethz.ch/kowalski/2013/05/21/bounded-gaps-between-primes/ _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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participants (6)
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Dan Asimov -
Fred lunnon -
rcs＠xmission.com -
Robert Baillie -
Veit Elser -
Victor Miller