TITLE: Solve: $ab+bc+ca\mid (a+b+c)^2$
QUESTION [6 upvotes]: I couldn't make any progress on this problem, can anyone help?
I found it's the same as:
Find all integers $a,b,c$ such that $ab+bc+ca$ divides $a^2+b^2+c^2$.
I found a solution $a=-b=1$, and $c$ any integer.
Any more solutions?

REPLY [4 votes]: Did not notice this question. The diophantine equation
$$ x^2 + y^2 + z^2 = B (yz + zx + xy)  $$
has  integer solutions $(x,y,z)$ not all equal to zero (and allowed negative)), if any only if we may express both
$$ B-1 = u_1^2 + 3 v_1^2, $$
$$ B+2 = u_2^2 + 3 v_2^2             $$
all in integers. The values that work are
$$ B = 1,2,5,10, 14,... $$
See
$x^2+y^2+z^2=5(xy+yz+zx)$ -- Is this all solutions? 
and my several answers at
Find a solution: $3(x^2+y^2+z^2)=10(xy+yz+zx)$ 
examples for $B = 5,10,14.$ For $B = 5$ we take the three values 
$$  (    5 u^2    + 9 uv  +    3 v^2,
      3   u^2  -3 uv    - v^2,
     -   u^2 +  uv    +  5 v^2 ).
$$
Next, take the triple in decreasing absolute value. Finally, if the first one is negative, negate all three. The result is a list that is not too repetitive, with $x \geq y \geq |z|,$ because $y$ turns out to be positive in this recipe. Sometimes $z$ is also positive, not often. Oh, nice rule, we get to take $u,v \geq 0.$
 ./isotropy_binaries_combined 1 5 300 | sort -n
              x      y      z                       u  v 
              5      3     -1      < 5, 9, 3 >      1  0    
             17      5     -1      < 5, 9, 3 >      1  1    
             41      5      3      < 5, 9, 3 >      2  1    
             59     47    -15      < 5, 9, 3 >      1  3    
             75     17     -1      < 5, 9, 3 >      3  1    
             89     83    -25      < 5, 9, 3 >      1  4    
            101     47    -15      < 5, 9, 3 >      2  3    
            111     17      5      < 5, 9, 3 >      3  2    
            129    125    -37      < 5, 9, 3 >      1  5    
            173     59    -15      < 5, 9, 3 >      5  1    
            185    131    -43      < 5, 9, 3 >      2  5    
            185    167    -51      < 5, 9, 3 >      1  6    
            201     83    -25      < 5, 9, 3 >      3  4    
            215     41      3      < 5, 9, 3 >      4  3    
            227     41      5      < 5, 9, 3 >      5  2    
            237     89    -25      < 5, 9, 3 >      6  1    
            251    215    -67      < 5, 9, 3 >      1  7    
            255    131    -43      < 5, 9, 3 >      3  5    
            293    255    -79      < 5, 9, 3 >      2  7 
              x      y      z                       u  v 

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For $B=10,$ we take
$$  (    5 u^2    + 8 uv  +    2 v^2,
  2   u^2  -4 uv    - v^2,
 -   u^2 + 2 uv    +  5 v^2 ).
$$
 ./isotropy_binaries_combined 1 10 300 | sort -n 
              x      y      z                       u  v 
              5      2     -1      < 5, 8, 2 >      1  0    
             29     23    -10      < 5, 8, 2 >      1  2    
             38      5     -1      < 5, 8, 2 >      2  1    
             50     47    -19      < 5, 8, 2 >      1  3    
             71      5      2      < 5, 8, 2 >      3  1    
             86     53    -25      < 5, 8, 2 >      2  3    
            101     23    -10      < 5, 8, 2 >      3  2    
            134     95    -43      < 5, 8, 2 >      1  5    
            167     29    -10      < 5, 8, 2 >      5  1    
            173     95    -46      < 5, 8, 2 >      3  4    
            191    125    -58      < 5, 8, 2 >      1  6    
            194     53    -25      < 5, 8, 2 >      4  3    
            215    146    -67      < 5, 8, 2 >      3  5    
            230     47    -19      < 5, 8, 2 >      6  1    
            263     50    -19      < 5, 8, 2 >      5  3    
            269    230    -97      < 5, 8, 2 >      2  7    
            290    149    -73      < 5, 8, 2 >      4  5 
              x      y      z                       u  v 

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For $B=14,$ we take
$$  (    3 u^2    + 6 uv  +    2 v^2,
  2   u^2  -2 uv    - v^2,
 -   u^2     +  3 v^2 ).
$$
No $uv$ term in the third form. Go Figure. 
 ./isotropy_binaries_combined 1 14 300 | sort -n 
              x      y      z                       u  v 
              3      2     -1      < 3, 6, 2 >      1  0    
             11      2     -1      < 3, 6, 2 >      1  1    
             23     11     -6      < 3, 6, 2 >      1  2    
             26      3     -1      < 3, 6, 2 >      2  1    
             47     11     -6      < 3, 6, 2 >      3  1    
             59     47    -22      < 3, 6, 2 >      1  4    
             66     23    -13      < 3, 6, 2 >      2  3    
             71      3      2      < 3, 6, 2 >      3  2    
             74     23    -13      < 3, 6, 2 >      4  1    
             83     74    -33      < 3, 6, 2 >      1  5    
            107     39    -22      < 3, 6, 2 >      5  1    
            111    107    -46      < 3, 6, 2 >      1  6    
            122     71    -37      < 3, 6, 2 >      2  5    
            131     39    -22      < 3, 6, 2 >      3  4    
            138     11     -1      < 3, 6, 2 >      4  3    
            146    143    -61      < 3, 6, 2 >      1  7    
            146     59    -33      < 3, 6, 2 >      6  1    
            167     66    -37      < 3, 6, 2 >      3  5    
            183     11      2      < 3, 6, 2 >      5  3    
            191    179    -78      < 3, 6, 2 >      1  8    
            191     83    -46      < 3, 6, 2 >      7  1    
            194    143    -69      < 3, 6, 2 >      2  7    
            218     59    -33      < 3, 6, 2 >      4  5    
            227     23     -6      < 3, 6, 2 >      5  4    
            239     66    -37      < 3, 6, 2 >      7  2    
            242    111    -61      < 3, 6, 2 >      8  1    
            242    219    -97      < 3, 6, 2 >      1  9    
            251    138    -73      < 3, 6, 2 >      3  7    
            282    239   -109      < 3, 6, 2 >      2  9    
            291     47    -22      < 3, 6, 2 >      7  3    
            299    183    -94      < 3, 6, 2 >      3  8    
            299    263   -118      < 3, 6, 2 >      1  10  
              x      y      z                       u  v 

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For $B = 29$ we need two parametrizations;
$$ 
( 23 u^2  +  49 u v +    17  v^2,
 17  u^2  -15  u v   -9  v^2,
 -9  u^2   -3  u v  + 23  v^2)$$
$$    ( 27  u^2  + 45  u v +  11 v^2,
 11  u^2  -23  u v   -7 v^2,
 -7  u^2  +  9  u v +  27 v^2) $$
 ./isotropy_binaries_combined 1 29 1111 | sort -n
              x      y      z                          u  v 
             23     17     -9      < 23, 49, 17 >      1  0    
             27     11     -7      < 27, 45, 11 >      1  0    
             83     29    -19      < 27, 45, 11 >      1  1    
             89     11     -7      < 23, 49, 17 >      1  1    
            207     29    -19      < 23, 49, 17 >      2  1    
            209     17     -9      < 27, 45, 11 >      2  1    
            263    261   -121      < 27, 45, 11 >      1  3    
            323    189   -109      < 23, 49, 17 >      1  3    
            371     99    -67      < 23, 49, 17 >      3  1    
            389     23     -9      < 27, 45, 11 >      3  1    
            461    383   -193      < 27, 45, 11 >      1  4    
            477    269   -157      < 27, 45, 11 >      2  3    
            491    347   -187      < 23, 49, 17 >      1  4    
            539    153   -103      < 23, 49, 17 >      2  3    
            557     99    -67      < 27, 45, 11 >      3  2    
            569     27     -7      < 23, 49, 17 >      3  2    
            693    551   -283      < 23, 49, 17 >      1  5    
            833    737   -361      < 27, 45, 11 >      2  5    
            911    153   -103      < 27, 45, 11 >      5  1    
            929    801   -397      < 23, 49, 17 >      1  6    
            959    477   -289      < 27, 45, 11 >      3  4    
           1007    509   -307      < 23, 49, 17 >      2  5    
           1019    693   -379      < 27, 45, 11 >      1  6    
           1067    251   -171      < 23, 49, 17 >      3  4    
           1071    239   -163      < 27, 45, 11 >      4  3 
           1109     27     11      < 23, 49, 17 >      4  3   
              x      y      z                          u  v