How many possible values of $k$ are there?

Question

I considered this problem, but, I cannot solve.

$a,b,c$ are a real numbers with $(a,b,c)\neq (0,0,0)$.

$k$ is defined as follows. $$k=\frac{(a^2+b^2+c^2)(a^3+b^3+c^3)}{(a^5+b^5+c^5)}$$ if $ab+bc+ca=0$, How many possible values of $k$ are there?

Firstly, I substitute $c=-\frac{ab}{a+b}$ into the equation of $k$, when the value of a+b is not $0$. Then,I got this equation.

$$k=1-\frac{2 a^8 b^2 + 8 a^7 b^3 + 16 a^6 b^4 + 20 a^5 b^5 + 16 a^4 b^6 + 8 a^3 b^7 + 2 a^2 b^8}{a^{10} + 5 a^9 b + 10 a^8 b^2 + 10 a^7 b^3 + 5 a^6 b^4 + a^5 b^5 + 5 a^4 b^6 + 10 a^3 b^7 + 10 a^2 b^8 + 5 a b^9 + b^{10}}=$$

However, this equation is too messy.

Secondly,I made an elementary symmetric polynomial from the equation of $k$. Then, I got this equation. $$k=\frac{(a+b+c)^3+3abc}{(a+b+c)^3+5abc}$$ However, I cannot proceed from here.

If you can solve this, could you tell me the answer or the hints? I'm sorry my broken English,I'm Japanese.

Answer 1

Looking at integers only. One infinite family is $$ a = u^2 - v^2 \; , \; \; \; b = 2(uv+v^2) \; , \; \; c = 2(-uv+v^2) \; , \; \; $$

Your ratio, fifth powers in the denominator, comes out $$ \frac{u^6 - 3 u^4 v^2 + 51 u^2 v^2 + 15 v^6}{u^6 - 11 u^4 v^2 + 67 u^2 v^4 + 7 v^6} $$

This is homogeneous, so we may divide through by $v^6,$ then draw a grph using $x = \frac{u}{v}.$ allowing for irrational $x$ gives us all values from $1$ to $15/7.$ Out of these there are infinitely many rational values when $u,v$ go back to being integers

Some output, integer pairs $(u,v)$ that are coprime with $u+v$ odd.

 u: 0  v: 1   numer: 15 =  3 5   denom: 7 =  7 
 u: 1  v: 0   numer: 1 =   1    denom: 1 =   1  
 u: 1  v: 2   numer: 1765 =  5 353   denom: 1477 =  7 211 
 u: 1  v: 4   numer: 74449 =  74449   denom: 45649 =  191 239 
 u: 2  v: 1   numer: 235 =  5 47   denom: 163 =  163 
 u: 2  v: 3   numer: 27091 =  27091   denom: 25291 =  7 3613 
 u: 2  v: 5   numer: 360739 =  151 2389   denom: 272539 =  272539 
 u: 3  v: 2   numer: 8061 =  3 2687   denom: 7261 =  53 137 
 u: 3  v: 4   numer: 175785 =  3 5 11719   denom: 169513 =  179 947 
 u: 4  v: 1   numer: 4159 =  4159   denom: 2359 =  7 337 
 u: 4  v: 3   numer: 74215 =  5 14843   denom: 70687 =  70687 
 u: 4  v: 5   numer: 729271 =  729271   denom: 713071 =  499 1429 
 u: 5  v: 2   numer: 29485 =  5 5897   denom: 15373 =  15373 
 u: 5  v: 4   numer: 373465 =  5 113 661   denom: 363097 =  7 51871 
 u: 6  v: 1   numer: 44619 =  3 107 139   denom: 34819 =  34819 
 u: 7  v: 2   numer: 129781 =  233 557   denom: 64981 =  7 9283 
 u: 7  v: 4   numer: 703585 =  5 140717   denom: 564193 =  7 80599 
 u: 8  v: 1   numer: 253135 =  5 50627   denom: 221383 =  107 2069 
 u: 8  v: 3   numer: 426871 =  426871   denom: 209071 =  209071 
 u: 9  v: 2   numer: 519765 =  3 5 34651   denom: 330037 =  330037