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If a and b are integers not both equal to 0, then there exist integers u and v such that GCD(a,b)=au+bv, where GCD(a,b) is the greatest common divisor of a and b.

Let beta=detB=x^2-ty^2, (1) where B is the Brahmagupta matrix, then det[B(x_1,y_1) B(x_2,y_2)] = det[B(x_1,y_1)]det[B(x_2,y_2)] (2) = beta_1beta_2]. (3)

B(x,y)=[x y; +/-ty +/-x]. (1) It satisfies B(x_1,y_1)B(x_2,y_2)=B(x_1x_2+/-ty_1y_2,x_1y_2+/-y_1x_2). (2) Powers of the matrix are defined by B^n = [x y; ty x]^n (3) = [x_n ...

If P(x) is an irreducible cubic polynomial all of whose roots are real, then to obtain them by radicals, you must take roots of nonreal numbers at some point.

product_(k=1)^(n)(1+yq^k) = sum_(m=0)^(n)y^mq^(m(m+1)/2)[n; m]_q (1) = sum_(m=0)^(n)y^mq^(m(m+1)/2)((q)_n)/((q)_m(q)_(n-m)), (2) where [n; m]_q is a q-binomial coefficient.

A variable that may assume complex values.

A map f:X-->Y is called constant with constant value y if f(x)=y for all x in X, i.e., if all elements of X are sent to same element y of Y.

A number taken to the power 3 is said to be cubed, so x^3 is called "x cubed." This terminology derives from the fact that the volume of a cube of edge length x is given by ...

int_0^inftye^(-omegaT)cos(omegat)domega=T/(t^2+T^2), which can be computed using integration by parts.

Given a Jacobi amplitude phi and a elliptic modulus m in an elliptic integral, Delta(phi)=sqrt(1-msin^2phi).