a/b + b/c + c/a = n

Sun Jul 27 10:34:01 CEST 2003

SeqFans,

Dave Rusin has now posted an exhaustive treatment of this topic
in a post to sci.math (cited below). The relevant sequences are:
http://www.research.att.com/projects/OEIS?Anum=A072716 and
http://www.research.att.com/projects/OEIS?Anum=A085705
%S A085705
3,5,6,9,10,13,14,15,16,17,18,19,20,21,26,29,30,31,35,36,38,40,41,44,47,
%T A085705 51,53,54,57
%N A085705 Integers expressible as (x^3 + y^3 + z^3)/(x*y*z) with
non-zero integers x, y, and z. (Alternatively, the in\
  tegers expressible as x/y + y/z + z/x with non-zero integers x, y, and
z.).

which I had submitted after running a search for one week cpu time
admitting also negative a,b,c.

Both sequences can now be extended.

Hugo Pfoertner

 Author: Dave Rusin <rusin at vesuvius.math.niu.edu> 
Date Posted: Jul 25 2003 9:03:33:000PM

In article <3F180917.CE3007B4 at abouthugo.de>,
Hugo Pfoertner  <nothing at abouthugo.de> wrote about the Diophantine
problem in the subject line, quoting a message in The Math Forum
citing this reference:
>TWO MORE REPRESENTATION PROBLEMS
>by Andrew Bremner and Richard Guy
>published in Proc. Edinburgh Math. Soc. vol 40 1997 pp 1-17.

For anyone interested in this problem: that's a fine paper to read;
it includes quite a references to the literature, tables of 
numerical solutions, and a discussion of the elliptic-curve techniques
needed to carry out the computations.

But as they say, "a week in the laboratory can frequently save an
hour's trip to the library!". I worked out most of the details myself
and had access to a bundle of computers, so I have quite a complete
set of data now. I'm not going to post it all to the newsgroup,
but I will make it available at
http://www.math.niu.edu/~rusin/research-math/abcn/

Here are some excerpts:
The following is the complete set of the 111 values of n under 200 for 
which the equation x^3+y^3+z^3=nxyz  can be solved in nonzero integers:

3, 5, 6, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 26, 29, 30, 31,
35, 36, 38, 40, 41, 44, 47, 51, 53, 54, 57, 62, 63, 64, 66, 67, 69,
70, 71, 72, 73, 74, 76, 77, 83, 84, 86, 87, 92, 94, 96, 98, 99, 101,
102, 103, 105, 106, 107, 108, 109, 110, 112, 113, 116, 117, 119, 120,
122, 123, 124, 126, 127, 128, 129, 130, 132, 133, 136, 142, 143, 145,
147, 148, 149, 151, 154, 155, 156, 158, 159, 160, 161, 162, 164, 166,
167, 172, 174, 175, 177, 178, 181, 185, 186, 187, 189, 190, 191, 192,
195, 196, 197

Except for the case n=5, there are infinitely many solutions for each 
n.
Except for n=142 and n=177, we have explicit solutions.

The following is a complete list of the values of n under 200 for which
the equation x^3+y^3+z^3=nxyz  can be solved in POSITIVE integers:
3, 5, 6, 9, 10, 13, 14, 17, 18, 19, 21, 26, 29, 30, 38, 41, 51, 53, 54,
57, 66, 67, 69, 73, 74, 77, 83, 86, 94, 101, 102, 105, 106, 110, 113,
117,
122, 126, 129, 130, 133, 145, 147, 149, 154, 158, 161, 162, 166, 174,
178,
181, 186, 195, 197
except for the possibility that the values n=147 and n=177 must be
added.
(We can know that as soon as we have explicit solutions for these two n.
That, in turn, can be accomplished by sufficient computing power,
probably
to be measured in CPU-weeks.  N.B. -- We have already 
expended over one hundred CPU-days for this project!)

Pfoertner goes on to say:
>N = 62 and N = 64 both have solutions but they are quite large.

What, a dozen or two digits? Piffle!

The techniques we discuss show that the equation 
                             a/b + b/c + c/a = 112
can be solved with explicit integers  a,b,c; what appears to be the
simplest solution requires (90+)-digit numbers:

             a =  44488222032517984080347242004206223609176772084

4845203037340381653808676781078204185344064777425
             b =    180001063934056147663194703762128694791524068

4971323481294582383858472523311320365128373281158 
             c =     -1331809157685411330016283859165784168699351

9959959149070559988026538909081959649861205201860

dave