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

[Dean Hickerson]

I wrote:

> For each such triple, we get two triples (a,b,c):  (e^2 f, f^2 d, d^2 e)
> and  (e f^2, d f^2, e d^2).

Hans Havermann asked:

> Is that second triple correct?

Oops, no it's not.  It should be  (e f^2, f d^2, d e^2).

> Anyways, I did a quick implementation of this taking {d, e, f} up to 1000:
>
> {3, 5, 6, 9, 10, 13, 14, 17, 18, 19, 21, 26, 29, 30, 38, 41, 51, 53, 
> 54, 57, 66, 69, 83, 86, 94, 105, 106, 126, 149, 154, 161, 166, 174, 
> 178, 195, 201, 209, 230, 237, 243, 250, 261, 269, 294, 323, 326, 329, 
> 366, 405, 446, 451, 478, 489, 534, 581, 622, 629, 630, 633, 681, 726, 
> 734, 789, 846, 905, 966, 978, 1011, 1097, 1410, 1491, 1658, 1713, 1718, 
> 1725, 1769, 1875, 1893, 2163, 2309, 2369, 2378, 2681, 3974, 6318, 7061, 
> 10995, 13971, 14803, 18014, 20778, 21529, 59802}

My program has reached max(d,e,f)=5063, and found 109 more values, starting
with:

    74, 77, 101, 129, 147, 162, 339, 526, 542, 563, 570, 638, 659, 1029

I've temporarily placed my list of quadruples (d,e,f,n) at

    http://www.math.ucdavis.edu/~dean/defn.txt

in case anyone wants it.

> Is there a simple rule the relates the maximum {d, e, f} to how many of 
> the found terms might be sequential?

I don't know.

Dean Hickerson
dean at math.ucdavis.edu