Math Forum Date: Jul 12, 2003 5:00 PM From: haoyuep@aol.com (Dan Hoey) Subject: Re: Problem: a/b + b/c + c/a = n On 10 Jul 2003, Marcus wrote: > I'm stuck on this problem that was posed on another message board. > The problem is to find which positive integers n can be the result > of a/b + b/c + c/a, where a, b, and c are positive integers. > Marcus I don't know why this question hasn't shown up on Usenet or Google, but that may be why you've gotten so few answers. I think the problem is well-known (though not to me). Anyway, an approach is to assume gcd(a,b,c)=1, though a, b, and c may not be pairwise relatively prime. Then it turns out that we can find integers t,u,v that are pairwise relatively prime, and for which a=t^2 u, b=u^2 v, c=v^2 t. The same (t,u,v) also yields a triple a=t^2 v, b=v^2 u, c=u^2 t, which is different if the t,u,v are different (which can only occur with (1,1,1) and (1,1,2)). Anyway, n=(t^3+u^3+v^3)/(tuv). My program tries u=1,2,...,50 (though it could easily go further on a better Lisp). For each u it tries all 1 <= t <= u with gcd(t,u)=1. For each t and u it tries all divisors v|t^3+u^3 with v >= u. The result (sorted by n) is: n t u v 3 1 1 1 5 1 1 2 6 1 2 3 9 2 3 7 10 5 7 18 13 9 13 38 14 2 7 13 17 5 18 37 18 13 42 95 19 1 5 9 21 2 13 21 26 9 38 91 29 27 43 182 30 2 21 31 41 2 31 43 41 1 5 14 41 1 2 9 51 9 13 77 53 2 7 27 54 2 43 57 66 1 3 14 83 5 9 61 106 1 35 54 149 1 14 45 154 2 13 63 161 11 38 259 174 5 7 78 178 2 27 97 195 7 15 143 250 2 9 67 261 3 7 74 269 1 14 61 323 9 49 377 326 5 14 151 451 23 31 567 478 13 23 378 633 3 14 163 978 33 43 1178 978 2 7 117 1011 7 11 279 1410 3 22 305 1658 19 26 905 1718 9 19 542 1769 2 5 133 1875 15 19 731 1893 3 43 494 1914 31 45 1634 2163 1 49 325 2309 2 13 245 2369 1 9 146 2681 1 49 362 4018 14 45 1591 5246 13 31 1454 5430 19 25 1606 6638 26 37 2527 7061 2 35 703 8346 31 37 3094 10995 3 23 871 12105 7 18 1235 13971 3 11 679 14803 1 9 365 15722 7 38 2045 17859 37 45 5453 18014 1 35 794 20778 2 13 735 21529 1 14 549 24761 7 31 2318 31130 14 43 4329 33081 3 31 1754 39290 7 45 3518 59802 1 14 915 82949 5 13 2322 84178 9 19 3794 92454 5 39 4246 112901 5 34 4381 118366 2 31 2709 187001 2 13 2205 191974 9 46 8915 196026 7 26 5973 203654 2 43 4185 941491 1 45 6509 2000041 1 29 24390 Dan