Show that for all K,L there exists some pair (a,b) such that \(K^2+3L^2=a^2+b^2-ab\)

[KingKong3]

I'm stumped, this is my first experience with a proof like this. I tried many different things, but I was only able to prove it for certain K, L.

Additionally, I tried proving that for all (a,b) there exists some K,L that satisfies the equation, but I had no luck with that too.

 

[skeeter]

what do K and L represent?

 

[Deleted member 4993]

I'm stumped, this is my first experience with a proof like this. I tried many different things, but I was only able to prove it for certain K, L.

Additionally, I tried proving that for all (a,b) there exists some K,L that satisfies the equation, but I had no luck with that too.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

 

[Dr.Peterson]

Show that for all K,L there exists some pair (a,b) such that K²+3L²=a²+b²−ab

I'm stumped, this is my first experience with a proof like this. I tried many different things, but I was only able to prove it for certain K, L.

Additionally, I tried proving that for all (a,b) there exists some K,L that satisfies the equation, but I had no luck with that too.

If all four numbers can be any real number, then both the LHS and the RHS can be any positive number. So for any value of the LHS, there is a pair (a, b), in fact infinitely many of them, such that the RHS has that value. (For example, if a = b = 1, then the RHS is 1; multiply both a and b by the square root of whatever number you need to get.)

I suspect you omitted something; maybe all the numbers are integers?

We'd really like to see this question in its context.