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In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the degree.
That lemma says that if the polynomial factors in Q[X], then it also factors in Z[X] as a product of primitive polynomials. Now any rational root p / q corresponds to a factor of degree 1 in Q [ X ] of the polynomial, and its primitive representative is then qx − p , assuming that p and q are coprime.
The system + =, + = has exactly one solution: x = 1, y = 2 The nonlinear system + =, + = has the two solutions (x, y) = (1, 0) and (x, y) = (0, 1), while + + =, + + =, + + = has an infinite number of solutions because the third equation is the first equation plus twice the second one and hence contains no independent information; thus any value of z can be chosen and values of x and y can be ...
[2] [3] Thus, in the expression 1 + 2 × 3, the multiplication is performed before addition, and the expression has the value 1 + (2 × 3) = 7, and not (1 + 2) × 3 = 9. When exponents were introduced in the 16th and 17th centuries, they were given precedence over both addition and multiplication and placed as a superscript to the right of ...
This is the same as asking for all integer solutions to + =; any solution to the latter equation gives us a solution = /, = / to the former. It is also the same as asking for all points with rational coordinates on the curve described by x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} (a circle of radius 1 centered on the origin).
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.
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All cases of the form (2, 3, n) or (2, n, 3) have the solution 2 3 + 1 n = 3 2 which is referred below as the Catalan solution. The case x = y = z ≥ 3 is Fermat's Last Theorem , proven to have no solutions by Andrew Wiles in 1994.
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