Can This Mathematical Equation Be Solved for All Prime Numbers?

[mathador]

Any pointers and/or help with proving the following would be appreciated

For every prime number n, there exist positive integers (k,l,m) such that

k(m²-n²)=2(m³+n³-lm)

some examples (there could be several/many solutions for a given n)

{n,k,l,m}
{2, 14, 8, 2}
{3, 59, 264, 1}
{5, 53, 192, 3}
{7, 71, 264, 5},{7, 239, 6080, 1}
{11, 163, 1856, 5}

If the statement is true for prime n's, it can be shown to hold for composite n's as well.

Thanks, Mathador

 

[gcsetma]

Perhaps you can think of it as a linear problem in k and l. Given prime n see whether a specific m can be found such that a solution to the linear problem is positive.

As you may be aware, a trivial solution is (n,k, 2n², n) k=1,2,3,...

 

[ramsey2879]

Perhaps you can think of it as a linear problem in k and l. Given prime n see whether a specific m can be found such that a solution to the linear problem is positive.

As you may be aware, a trivial solution is (n,k, 2n², n) k=1,2,3,...

Also trival are solutions where m = 1