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Abel wrote: "The first and, if I am not mistaken, the only one who, before me, has sought to prove the impossibility of the algebraic solution of general equations is the mathematician Ruffini. But his memoir is so complicated that it is very difficult to determine the validity of his argument.
Proof. The equation A X + X B = C {\displaystyle AX+XB=C} is a linear system with m n {\displaystyle mn} unknowns and the same number of equations. Hence it is uniquely solvable for any given C {\displaystyle C} if and only if the homogeneous equation A X + X B = 0 {\displaystyle AX+XB=0} admits only the trivial solution 0 {\displaystyle 0} .
Any solutions to the Beal conjecture will necessarily involve three terms all of which are 3-powerful numbers, i.e. numbers where the exponent of every prime factor is at least three. It is known that there are an infinite number of such sums involving coprime 3-powerful numbers; [11] however, such sums are rare. The smallest two examples are:
The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables. This is the origin of the term linear for describing this type of equation.
It now follows from the properties of the Kronecker product that the equation AXB = C has a unique solution, if and only if A and B are invertible (Horn & Johnson 1991, Lemma 4.3.1). If X and C are row-ordered into the column vectors u and v , respectively, then ( Jain 1989 , 2.8 Block Matrices and Kronecker Products)
One of the widely used types of impossibility proof is proof by contradiction.In this type of proof, it is shown that if a proposition, such as a solution to a particular class of equations, is assumed to hold, then via deduction two mutually contradictory things can be shown to hold, such as a number being both even and odd or both negative and positive.
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]
Some sources allow convergence to 0 if there are only a finite number of zero factors and the product of the non-zero factors is non-zero, but for simplicity we will not allow that here. If the product converges, then the limit of the sequence a n as n increases without bound must be 1, while the converse is in general not true.