1 Bipartite Graphs Graphs that are 2-colorable are important enough to merit a special name; they are called bipartite graphs. Suppose that G is bipartite. Then we can color every vertex in G ei ther black or white so that adjacent vertices get different colors. Then we can put all the

1 RSA In 1977, Ronald Rivest, Adi Shamir, and Leonard Adleman proposed a highly secure cryp- tosystem(called RSa)based on number theory. Despite decades of attack, no significant weakness has been found (Well, none that you and me would know.)Moreover, RSA has a major advantage over traditional codes: the sender and receiver of an encrypted

1 The pulverizer We saw in lecture that the greatest common divisor(GCD)of two numbers can be written as a linear combination of them. That is, no matter which pair of integers a and b we are given, there is always a pair of integer coefficients s and t such that

Introduction to Probability Probability is the last topic in this course and perhaps the most important. Many Igorithms rely on randomization. Investigating their correctness and performance re- quires probability theory. Moreover, many aspects of computer systems, such as memory

Generating functions Generating functions are one of the most surprising, useful, and clever inventions in discrete math. Roughly speaking, generating functions transform problems about se- quences into problems about functions. This is great because weve got piles of mathe- matical machinery for manipulating functions. Thanks to generating functions, we can

Sums, Approximations, and Asymptotics II Block Stacking How far can a stack of identical blocks overhang the end of a table without toppling over? Can a block be suspended entirely beyond the table's edge? Table Physics imposes some constraints on the arrangement of the blocks. In particular, the stack falls off the desk if its center of mass lies beyond the desk's edge. Moreover, the center of mass of the top k blocks must lie above the(k+1)-st block;

Srini devadas and Eric Lehman Lecture notes Number theory ll Image of Alan Turing removed for copyright reasons s The man pictured above is Alan Turing, the most important figure in the history of mputer science. For decades, his

Number Theory I Number theory is the study of the integers. Number theory is right at the core of math ematics; even Ug the Caveman surely had some grasp of the integers- at least the posi tive ones. In fact, the integers are so elementary that one might ask, What's to study?

It's really sort of amazing that people manage to communicate in the English language Here are some typical sentences: 1. You may have cake or you may have ice cream 2. If pigs can fly, then you can understand the Chernoff bound 3. If you can solve any problem we come up with then you get an a for the course. 4. Every American has a dream What precisely do these sentences mean? Can you have both cake and ice cream or must you choose just one desert? If the second sentence is true, then is the Chernoff bound incomprehensible? If you can solve some problems we come up with but not all, then do you get an a for the course? And can you still get an a even if you cant solve any of the problems? Does the last sentence imply that all Americans have the same dream or might

1 Streaks someone tapping the H and t keys in a what felt like a random way?0 Nas the table of H's and T's below generated by flipping a fair coin 100 times