The inner feedback loop will be designed based on linear algebraic method, by solving a set of

Diophantine equations, while the outer loop will be designed using linear quadratic Gaussian (LQG) controller, which is one of the robust controllers.

We can show that this condition is equivalent to a = 2A/B with a rational solution (A, B) of the

Diophantine equation [A.sup.2] - 2[B.sup.2] = 1.

Particle swarm optimization based

diophantine equation solver.

The

Diophantine equation (11) determines only the polynomials R and S.

The

Diophantine equation has two solutions called solution 1 and solution 2.

The final theorem was not formulated all the time (not even with teachers), because other important aspects needed to be clarified (such as the sieve method for counting, or the properties of the linear

diophantine equation, including the existence of positive solutions), but all the activities had very substantial mathematical content, and this content was developed along the questions posed (or difficulties faced) by the participants.

Denoting by ([x.sub.v], [y.sub.v]) the v-th solution of the

Diophantine equation (2), ([x.sub.v], [y.sub.v]) satisfies the following recurrence relation:

(ii) It is known that the

Diophantine equation [x.sup.2] + [y.sup.2] = n has a nonnegative primitive solution if and only if every odd prime factor of n is congruent to 1 modulo 4 (see e.g.

(The same argument holds for Case 2.) Putting everything together, we get that the number of primes p [member of] (X/2, X) such that the

Diophantine equation (1) can have a non-trivial proper solution is

Hence we can write the equation 4n+8=2p as 4n - 2p= - 8 which is nothing but a

diophantine equation. There fore this equation has integer solution as g.c.d (4,-2) divides -8.

We search for nonzero integral triples (x, y, z) satisfying the quartic

Diophantine equation 2[x.sup.2][z.sup.2] = [y.sup.2]([x.sup.2] + [z.sup.2]).

It asks whether there is a mechanical procedure, such as could be programmed into a computer, for deciding whether a

Diophantine equation has solutions (a

Diophantine equation is one like the Fermat equation [x.sup.n] + [y.sup.n] = [Z.sup.n], where solutions must be whole numbers).