Search results
Results From The WOW.Com Content Network
With n, x, y, z ∈ N (meaning that n, x, y, z are all positive whole numbers) and n > 2, the equation x n + y n = z n has no solutions. Most popular treatments of the subject state it this way. It is also commonly stated over Z: [16] Equivalent statement 1: x n + y n = z n, where integer n ≥ 3, has no non-trivial solutions x, y, z ∈ Z.
As shown below, his proof is equivalent to demonstrating that the equation x 4 − y 4 = z 2. has no primitive solutions in integers (no pairwise coprime solutions). In turn, this is sufficient to prove Fermat's Last Theorem for the case n = 4, since the equation a 4 + b 4 = c 4 can be written as c 4 − b 4 = (a 2) 2.
It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions. [ 3 ] [ 4 ] Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more natural numbers , it would necessarily imply that a second ...
The equation + = has no solutions in positive integers and pairwise coprime integers A, B, C if x, y, z ≥ 2. The conjecture was formulated in 1993 by Andrew Beal , a banker and amateur mathematician , while investigating generalizations of Fermat's Last Theorem .
In general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an underdetermined system. In general, a system with the same number of equations and unknowns has a single unique solution. In general, a system with more equations than unknowns has no solution.
Vertical line of equation x = a Horizontal line of equation y = b. Each solution (x, y) of a linear equation + + = may be viewed as the Cartesian coordinates of a point in the Euclidean plane. With this interpretation, all solutions of the equation form a line, provided that a and b are not both zero. Conversely, every line is the set of all ...
One particular solution is x = 0, y = 0, z = 0. Two other solutions are x = 3, y = 6, z = 1, and x = 8, y = 9, z = 2. There is a unique plane in three-dimensional space which passes through the three points with these coordinates, and this plane is the set of all points whose coordinates are solutions of the equation.
Therefore, the solution = is extraneous and not valid, and the original equation has no solution. For this specific example, it could be recognized that (for the value x = − 2 {\displaystyle x=-2} ), the operation of multiplying by ( x − 2 ) ( x + 2 ) {\displaystyle (x-2)(x+2)} would be a multiplication by zero.