From philomath@voicenet.comThu Jul 18 19:28:38 1996
Date: Wed, 17 Jul 1996 08:55:22 -0400
From: Scott Aaronson <philomath@voicenet.com>
To: Multiple recipients of list NMBRTHRY <NMBRTHRY@listserv.nodak.edu>
Subject: Help with Diophantine equation

Hello number theorists!  I have a problem that you might be able to help
me with.  Consider the Diophantine equation

   n
 _____
 \   /  1      1
  >    ____ = ____
 /___\ a(i)    k
  i=0

with n, k, and a(0..n) integers > 0.  I have observed that, for n held
constant, the amount of unordered solution sets for a increases
generally (but not uniformly) as k increases. My questions are:

1) Can the amount of solutions for a be generalized in terms of k when n
equals, say, 2?
2) Can the amount of solutions for a be generalized in terms of (k,n)?

(You may note that, when k=n=2, solving for a is equivalent to solving
the trivial problem "Find all cuboids of integral side such that the
surface area (in square units) equals the volume (in cubic units)").

For n=2, here is the amount of solutions for 0<k<25:

k       # of unordered a
1       3
2       10
3       21
4       28
5       36
6       57
7       42
8       70
9       79
10      96
11      62
12      160
13      59
14      136
15      196
16      128
17      73
18      211
19      80
20      292
21      245
22      157
23      93
24      366

Any help would be appreciated.
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