[SOLVED] algorithm math Go Homework #3, Math 440/540

Homework #3, Math 440/540
You may be as sophisticated as you wish in your Mathematics usage, unless otherwise noted. However, include brief comments in your code so that a fellow student would be able to follow your work.
1.) Write a Mathematica procedure implementing the Extended Eu- clidean Algorithm. Show that this works by displaying various appli- cations of your procedure with explicit numbers.
2.) By Theorem 3.23 of the text, the linear diophantine equation of the form ax + by = c has no integral solutions if c is not divisible by (a, b), the greatest common divisor of a and b. On the other hand if (a, b) divides c, then we can use the Extended Euclidean Algorithm to find integers s, t such that sa + tb = (a, b); multiplying through by the correct factor gives an integral solution x, y.
Write a Mathematica procedure that solves any linear diophantine equation of the form ax + by = c, whenever it is solvable. You should invoke your Extended Euclidean Algorithm.
3.) The proof of Theorem 12.8 of the text shows that every rational number can be written as a finite simple continued fraction. Indeed, each rational number is of the form a/b with b > 0. Thus, letting r0 = a, r1 = b, the Euclidean Algorithm gives a sequence of equations, r0 =r1q1 +r2,r1 =r2q2 +r3,. Thus,
a=q+r2=q+ 1 =q+ 1 ==q+ 1 b1r1r/r1r31 1
.
1 1 2 q2+r2 q2++ 1
1
qn1 + qn
The final displayed quantity becomes so unwieldy that we define in general
[a0;a1,,ak] = a0 +
1
a1 +
+
ak1 + ak 1
.
1
1
1

2
The continued fraction convergents of such a continued fraction are the k + 1 rational numbers
a0,[a0;a1]=a0 +1/a1,,[a0;a1,,ak].
Write a Mathematica procedure to find the continued fraction con-
vergents of any given rational number.
4.) Other resources: Open a web browser, to
http://www.ams.org/mathscinet/search
You will need to do this either on campus, or with a browser which invokes the OSU library proxy.
In the Title box, type Chinese remainder theorem, then click on Start Search.
a.) How many articles with these words in the title are reviewed?
b.) Go back to the search page. Find the review of an article entitled A centennial history of the prime number theorem. What is the final sentence of the review?
c.) Go back to the search page. Delete the title entry, and insert rosen, k* in the author field. Find the review of the fourth edition of our text. What is the first sentence of the review?
d.) Now choose your own topic and use the Search to find out something of interest. Explain.