# Can This Mathematical Equation Be Solved for All Prime Numbers?

In summary, the conversation is discussing a statement that states for every prime number n, there exist positive integers (k,l,m) that satisfy the equation k(m2-n2)=2(m3+n3-lm). The conversation also provides some examples of solutions for specific values of n, and mentions that the statement may also hold for composite numbers. Additionally, it is suggested to think of the problem as a linear problem in k and l, and a possible solution for n is (n,k,2n2,n).
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.

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,...

gcsetma said:
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

## 1. What is the equation K(m^2-n^2) = 2(m^3+n^3-lm)?

The equation K(m^2-n^2) = 2(m^3+n^3-lm) is a mathematical expression that relates three variables: K, m, and n. It states that the product of K and the difference between m squared and n squared is equal to twice the sum of m cubed, n cubed, and the product of l and m.

## 2. What is the significance of this equation?

This equation is significant because it is a polynomial equation of degree 3, also known as a cubic equation. It is used in various fields of science and engineering, such as physics, chemistry, and fluid dynamics, to model and solve complex problems and systems.

## 3. How is this equation derived?

This equation can be derived using algebraic manipulation and substitution. It can also be derived from the binomial theorem, which expands (m+n)^3 and (m-n)^3 to obtain the terms in the equation.

## 4. What are the possible solutions to this equation?

As a cubic equation, this equation can have up to three possible solutions, depending on the values of K, m, n, and l. These solutions may be real or complex numbers, and can be determined using various methods such as factoring, completing the square, or using the cubic formula.

## 5. How is this equation used in scientific research?

This equation is used in scientific research to model and solve various problems and systems. For example, it can be used to calculate the volume of a cube, the flow rate of a fluid, or the energy of a chemical reaction. It can also be used in data analysis and curve fitting to determine relationships between variables.

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