# 250 Problems In Elementary Number Theory

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## 250 problems in elementary number theory

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More geometry advice: Do lots of locus problems to build up your geometric intuition. Avoid the cheap tricks (such as converting everything to algebra, overuse of trigonometry, "complex numbers", "barycentric coordinates", "apply a projective transformation to send a line to infinity", "invert the diagram to turn circles into lines", "it's enough to check the statement holds at three points", etc.), try to learn the underlying philosophy and intuition behind geometry properly. Never apply a symmetry to an entire diagram, the correct application of symmetry relates points to other points on the same diagram (examples: the Fermat point, Fagnano's problem, the Euler line, most applications of spiral similarity, perspectivity/cross ratio chase arguments such as the proof of Pappus's Theorem or Desargues's Theorem; nonexample: the way most people try to apply inversion).Algebra and Inequalities: "A

Lecture, discussion. Basic concepts of discrete mathematics useful to computer science: set theory, strings and formal languages, propositional and predicate calculus, relations and functions, basic number theory. Induction and recursion: interplay of inductive definition, inductive proof, and recursive algorithms. Elementary combinatorics. Basic graph theory. Finite-state machines, regular languages, nondeterministic finite automata, Kleene's Theorem. Problem sets, 2 midterm exams, timed final. Corequisite: MATH 132/136 or equivalent. Prerequisite: MATH 131 and (CMPSCI 187 or ECE 242). CS ENG majors may take CMPSCI 250 with ECE 242. Contact your Department Advisor. 4 credits.

Applied cryptography spans many disciplines including computer systems, computation, mathematics, and law. This experimental course aims to teach students both the theoretical foundations of cryptography and the humility of building practical cryptographicsystems. Topics will include computational indistinguishability, cryptographic attacks, applications of cryptography, and advancedtopics in block ciphers, hash functions, and public key cryptography. Students will be evaluated based on final group projects, classparticipation, problem sets with hands-on labs, and quizzes. Prerequisites: CMPSCI 311 and CMPSCI 377. Prior experience with number theory is recommended. 3 credits.

An in-depth introduction to the main models and concepts of the mathematical theory of computation, including: Computability: What problems can be solved in principle? How might you prove that a problem can't be solved? Complexity: What problems can be solved within given resource constraints? How do constraints on different resources (e.g., time, space, or parallel time) relate? Logic: What are the best ways to formally specify a problem, and how do these specifications relate to the difficulty of the problem? Prerequisites: Undergraduate-level courses in discrete mathematics (e.g., CMPSCI 250) and analysis of algorithms (e.g., CMPSCI 311 or better yet 611), plus additional mathematical maturity (e.g., CMPSCI 611, A's in 250 or 311, or other mathematical background). Previous background in formal languages (e.g., CMPSCI 401) is quite desirable. Course requirements: about eight problem sets, timed midterm and final. Also open to qualified undergraduates. 3 credits.

Lecture, discussion. Basic concepts of discrete mathematics useful to computer science: set theory, strings and formal languages, propositional and predicate calculus, relations and functions, basic number theory. Induction and recursion: interplay of inductive definition, inductive proof, and recursive algorithms. Elementary combinatorics. Basic graph theory. Finite-state machines and regular languages: Myhill-Nerode Theorem, nondeterministic finite automata, Kleene's Theorem. Problem sets, 2-3 midterm exams, timed final. Corequisite: MATH 132/136 or equivalent. Prerequisite: CMPSCI 187 or equivalent. 4 credits.

The mathematical classification of computational problems as solvable or unsolvable, and as easy or difficult. Review of finite state machines and regular languages. Introduction to context-free languages and pushdown automata. General models of computation, provably unsolvable problems. Resource-bounded computation, the classes P and NP, further topics in complexity theory such as interactive proofs. Problem sets, midterm, final, some group activities. Prerequisites: CMPSCI 250 and 311 or permission of instructor. 3 credits.

An in-depth introduction to the main models and concepts of the mathematical theory of computation, including: Computability: What problems can be solved in principle? How might you prove that a problem can't be solved? Complexity: What problems can be solved within given resource constraints? How do constraints on different resources (e.g., time, space, or parallel time) relate? Logic: What are the best ways to formally specify a problem, and how do these specifications relate to the difficulty of the problem? Text: Papadimitriou, "Computational Complexity". Prerequisites: Undergraduate-level courses in discrete mathematics (e.g., CMPSCI 250) and analysis of algorithms (e.g., CMPSCI 311 or better yet 611), plus additional mathematical maturity (e.g., CMPSCI 611, A's in 250 or 311, or other mathematical background). Previous background in formal languages (e.g., CMPSCI 401) useful but not required -- students with none may wish to consult Lewis and Papadimitriou, "Elements of the Theory of Computation", or Hopcroft and Ullman, "Introduction to Automata Theory, Languages, and Computation". Course requirements: about six problem sets, timed midterm and final. Also open to qualified undergraduates. 3 credits.

The study of the resources required to solve different problems in various abstract models of computation. Sequential computation: Turing machines, non-determinism, alternation, algebraic automaton theory. Parallel computation: Boolean circuits, branching programs, uniformity. Lower bounds for circuit models. Descriptive Complexity. Possible optional topics depending on student interest: Approximation of NP-complete prob- lems, interactive proofs, non-uniform finite automata, dynamic complexity. Prerequisite: CMPSCI 601 or permission of instructor. Intended for inter- ested general graduate students as well as those concentrating in theory. Course format: Probably 2-3 problem sets, class presentation (may be survey or actual research, might lead to a synthesis paper). Textbook: Xeroxed notes and research papers, plus sections of Prof. Immerman's new book, "Descriptive Complexity". 3 credits.

We study more advanced material in the theory and implementation of algorithms. Possible topics include approximate string matching, computational biology, computational geometry, cryptography, approximate solution to combinatorial optimization problems, advanced data structures, parallel algorithms, and data compression. Typically, students research an area of mutual interest, and present it to the rest of the class. 1 credit. 041b061a72