question about parallel sequence making prime
Jeffrey Shallit
elvis at graceland.math.uwaterloo.ca
Thu Feb 22 15:21:15 CET 2007
such an integer exists. To do so, use the Chinese Remainder theorem
t == -b(1) (mod 2)
t == -b(2) (mod 3)
...
t == -b(k) (mod p_k)
such that b(1)+t, ..., b(k-1)+t was composite, it follows that a(k+1) >=
say q_1, q_2, ..., q_{k+1}.
t == q_1-a(1) (mod q_1^2)
t == q_2-a(2) (mod q_2^2)
...
t == q_2-a(k) (mod q_k^2)
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From: Hans Havermann <pxp at rogers.com>
Subject: Re: Lattice Animals (Island Reference)
Date: Thu, 22 Feb 2007 11:16:32 -0500
To: seqfan at ext.jussieu.fr
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Date: Thu, 22 Feb 2007 09:18:10 -0800
From: "Jonathan Post" <jvospost3 at gmail.com>
To: "franktaw at netscape.net" <franktaw at netscape.net>
Subject: Re: Lattice Animals with self-avoiding perimeters
Cc: seqfan at ext.jussieu.fr, "Forrest Bishop" <forrestb at ix.netcom.com>,
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Dear Frank,
True, but incomplete. Holes need not be square, let alone 1x1 monominoes.
There is a 13-omino which is the smallest lattice animal with
self-avoiding perimeter to have two holes (both monomino). Total
perimeter = 16 + 2*4 = 24. Topology is rooted tree with two vertices
each connected to the root.
There is a 18-omino which is the smallest lattice animal with
self-avoiding perimeter to have three holes (all monomino). Total
perimeter = 20 + 3*4 = 32. Topology is rooted tree with 3 vertices
each connected to the root.
There is a 21-omino which is the smallest lattice animal with
self-avoiding perimeter to have 4 holes (all monomino). Total
perimeter = 20 + 4*4 = 36. Topology is rooted tree with 4 vertices
each connected to the root.
These, above, are in your subset. But also:
There is a 16-omino which is the smallest lattice animal with
self-avoiding perimeter to have a square hole (3x3) which itself
contains a hole (monomino). Total perimeter = 20 + 12 + 4 = 36.
Topology is rooted tree with leaf vertex connected to middle vertex
connected to the root.
There is a 24-omino which is the smallest lattice animal with
self-avoiding perimeter to have a square hole (5x5) which contains an
island (3x3) which itself contains a hole (monomino). Total perimeter
= 28 + 20 + 12 + 4 = 64. Topology is rooted path of 4 vertices.
I misspoke in saying that these are homomorphic into trees. I mean
rooted trees.
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