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v. t. e. Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." [ 1 ]
The intersection point is the solution. In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. [1][2] For example, is a system of three equations in the three variables x, y, z. A solution to a linear system is an assignment of values to the variables such ...
The homogeneous (with all constant terms equal to zero) underdetermined linear system always has non-trivial solutions (in addition to the trivial solution where all the unknowns are zero). There are an infinity of such solutions, which form a vector space , whose dimension is the difference between the number of unknowns and the rank of the ...
Triviality (mathematics) In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). [1][2] The noun triviality usually refers to a simple technical aspect of some proof or definition.
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown (s) consists of one (or more) function (s) and involves the derivatives of those functions. [ 1 ] The term "ordinary" is used in contrast with partial differential equations (PDEs ...
In the following Diophantine equations, w, x, y, and z are the unknowns and the other letters are given constants: a x + b y = c {\displaystyle ax+by=c} This is a linear Diophantine equation or Bézout's identity. w 3 + x 3 = y 3 + z 3 {\displaystyle w^ {3}+x^ {3}=y^ {3}+z^ {3}} The smallest nontrivial solution in positive integers is 123 + 13 ...
If the matrix on the left is invertible, the unique solution is the trivial solution (A 1, A 2) = (x 1, x 2) = (0, 0). The non trivial solutions are to be found for those values of ω whereby the matrix on the left is singular; i.e. is not invertible. It follows that the determinant of the matrix must be equal to 0, so:
The Fredholm alternative is the statement that, for every non-zero fixed complex number either the first equation has a non-trivial solution, or the second equation has a solution for all . A sufficient condition for this statement to be true is for to be square integrable on the rectangle (where a and/or b may be minus or plus infinity).