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

[Don McDonald]

dear seqfans,

I limited my search to 80>=a>b>c>0.

I then limited solutions to
80>=a >b>=1,  a>=c>=1, gcd(a,b,c) ==1.   Exhaustive.

and found 3 primitive solutions. [I should have gone to a=81.]


don.mcdonald
12.07.03  22:35    
.. .Calc.Profile.eisintegsq.Seqfan.abc-n.abcn


          GP/PARI CALCULATOR Version 1.36
                 (Archimedes version)

    
? ?gcd
gcd(x,y)=greatest common divisor of x and y.

? #
    timer on

since Hugo , Marcus have more (higher) solutions
my method may be too slow.

? for(a=1,28,for(b=1,a-1,g=gcd(a,b);x=a/b;for(c=1,a,if(gcd(g,c)==1,n=x+b/c+c/a;if(denom(n)==1, print(a,"  ",b,"  ",c,"  ",n),),))))

a  b  c  n    primitive solutions.
== == == ==

4  1  2  5
12  9  2  6
18  4  3  6
time = 24,390 ms

? for(a=1,80,
(no more solutions.)
...

time = 10mn, 13,460 ms

          GP/PARI CALCULATOR Version 1.36
                 (Archimedes version)

? for(a=1,50,for(b=1,a-1,for(c=1,b-1,n=a/b+b/c+c/a;
if(denom(n)==1,print(a,"  ",b,"  ",c,"  ",n),))))

a   b  c  n    (includes only a>b>c and not primitive solutions.)
==  == == ==

12  9  2  6
18  4  3  6
24  18  4  6
36  8  6  6
36  27  6  6
48  36  8  6

? for(a=48,69,
...
48  36  8  6
54  12  9  6
60  45  10  6
? for(a=69,80,
...

72  16  12  6
72  54  12  6

>Hugo Pfoertner
      >Marcus