My suggestion is to take the whole thing a modulo 2 so you would have only a limited number of cases to check.lfdahl said:Find the pairs of nonnegative integers, $(m,n)$, which obey the equality:
\[(m-n)^2(n^2-m) = 4m^2n\]
So far, I haven´t found a single pair, but I cannot prove, that the set of solutions is empty.
Perhaps, someone can help me to crack this nut?
lfdahl said:Find the pairs of nonnegative integers, $(m,n)$, which obey the equality:
\[(m-n)^2(n^2-m) = 4m^2n\]
Not a solution but some elimination
So far, I haven´t found a single pair, but I cannot prove, that the set of solutions is empty.
Perhaps, someone can help me to crack this nut?
Try for a solution with $m=2n$: $$(2n-n)^2(n^2-2n) = 16n^3,$$ $$n^3(n-2) = 16n^3,$$ $$ n-2 = 16,$$ $$ n=18.$$ That gives the solution $(m,n) = (36,18).$lfdahl said:Find the pairs of nonnegative integers, $(m,n)$, which obey the equality:
\[(m-n)^2(n^2-m) = 4m^2n\]
So far, I haven´t found a single pair, but I cannot prove, that the set of solutions is empty.
Perhaps, someone can help me to crack this nut?
The purpose of finding pairs of integers (m,n) is to identify two numbers that satisfy a specific relationship or equation. This can be useful in various mathematical and scientific applications, such as solving equations, graphing functions, and analyzing data.
To find pairs of integers (m,n), you can use algebraic methods, such as substitution or elimination, to solve an equation or system of equations. You can also use visual methods, such as graphing, to identify points where the coordinates are both integers.
Yes, there can be multiple pairs of integers (m,n) that satisfy a given equation. For example, the equation y = 2x has an infinite number of solutions, where x and y are both integers.
Yes, there are several types of pairs of integers (m,n) that have specific properties or applications. Some examples include prime numbers, perfect squares, and Pythagorean triples, which have been studied extensively in number theory and have practical applications in cryptography and computer science.
Pairs of integers (m,n) are used in a wide range of real-world situations, from calculating distances and areas in geometry to predicting patterns and relationships in data. They are also commonly used in coding and programming to represent coordinates and indices in arrays and matrices.