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The fact that every polynomial equation of positive degree has solutions, possibly non-real, was asserted during the 17th century, but completely proved only at the beginning of the 19th century. This is the fundamental theorem of algebra , which does not provide any tool for computing exactly the solutions, although Newton's method allows ...
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.
When we allow the exponent n to be the reciprocal of an integer, i.e. n = 1/m for some integer m, we have the inverse Fermat equation a 1/m + b 1/m = c 1/m. All solutions of this equation were computed by Hendrik Lenstra in 1992. [169] In the case in which the mth roots are required to be real and positive, all solutions are given by [170]
If the rational root test finds no rational solutions, then the only way to express the solutions algebraically uses cube roots. But if the test finds a rational solution r, then factoring out (x – r) leaves a quadratic polynomial whose two roots, found with the quadratic formula, are the remaining two roots of the cubic, avoiding cube roots.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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.
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.
In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction [1] used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. [2]