I. Number and title of course

Mathematics 351-Elementary Theory of Numbers

A. This course is a course of enrichment material for prospective teachers.

B. The course will help students develop and improve their abilities to write mathematical proofs.

C. The course will broaden the background of students who are planning to do graduate work in mathematics.

D. The course will reinforce the students' ability to use the compute, in mathematical problem solving, a process that has been initiates in lower?level courses.

A. To provide students with the opportunity to use the computer to formulate conjectures and develop proofs through their investigations of number theoretic properties.

B. To develop the ability of the student to prove theorems about integers.

C. To explore the historical development of integer properties and the contributions of famous mathematician to number theory.

D. To inspire the student towards an involvement in the subject by considering some famous unsolved problems of number theory and by exploring the connections that number theory has with other branches of mathematics.

A. Divisibility theory in the integers

1. Definitions and elementary properties

2. The division algorithm

3. The greatest common divisor and least

4. The Euclidean algorithm

5. The Diophantine equation ax + by = c

6. Number theoretic functions

7. Euler's phi-functions
 
 

B. Prime numbers and their distribution

1. Definition

2. The fundamental theorem of arithmetic

3. The sieve of Eratosthenes and other methods of factoring

4. The Goldbach and other famous conjectures

5. Proofs of the infinitude of primes

6. The prime number theorem
 
 

C. The theory of congruences

1. Definitions and elementary properties

2. Special divisibility tests

3. Linear congruences

4. The Chinese remainder theorem

5. Fermat's little theorem

6. Wilson's theorem

7. Euler's theorem

D. Primitive roots and indices

1. The order of an integer module M

2. Primitive roots of primes

3. Composite numbers having primitive roots

4. The theory of indices

E. Other topics

1. Quadratic reciprocity law

2. The Legendre symbol and its properties

3. Perfect numbers

4. Mersenne primes

5. Fermat numbers

6. Pythagorean triples

7. Fermat's ''Last" theorem

8. Fibonacci sequences

9. Continued fractions

10. Pell's equation

Adams, W. and Goldstein, L., Introduction to Number Theory, Englewood Cliffs, NJ, Prentice Hall, 1980.

Andrews, George E., Number Theory, The Theory of Partitions, Reading, MA, Addison and Wesley, 1976.

Apostol, Tom, Introduction to Analytic Number Theory New York, NY, Springer Verlag, 1976.

Armendariz, Effraim P. and McAdams, Stephen Jr., Elementary Number Theory, New York, NY, Macmillan Publishing Co., 1980.

Baker, Alan, A Concise Introduction to the Theory of Numbers, New York, NY, Dover Publications Inc., 1964.

Beiler, Albert H., Recreations in the Theory of Numbers, New York, NY, Dover Publications Inc., 1964.

Burn, R.P. f A Pathway Into Number Theorem, New York, NY,Cambridge University Press, 1982.

Burton, Elementary Number Theory, Rockleigh, NJ, Allyn and Bacon, 1980.

Conway, John, On Numbers and Games, Orlando, FL, Academic Press, 1976.

Dickson, Leonard, E., History of the Theory of Numbers, New York, NY, Chelsea Publications, 1966, Carnegie Institute, 1979.

Dickson, Leonard E., Studies of the Theory of Numbers, New York, NY, Chelsea Publications, 1957.

Dudley, Underwood, Elementary Number Theory 2nd edition, San Francisco, CA, W.H. Freeman and Co., 1978.

Eynden, Charles V., Elementary Number Theory, New York, NY, Random House Inc., 1987.

Grosswald, Emil, Topics from the Theory of Numbers, Cambridge, MA, Birkhauser, 1982.

Gupta, H., Selected Topics in Number Theory, Abacus, England, 1980.

Hardy, Godfrey H. and Wright E.M., Introduction to the Theory of Numbers 5th edition, Fairlawn, NY, Oxford University Press, 1979.

Hua, L.K., Introduction to Number Theory, New York, NY,Springer Verlag, 1982.

Kohlitz, Neal, ea., Modern Trends in Number Theory Related to Fermat's Last Theorem, Cambridge MA, Berkhauser, 1982.

Landau, Edmund, Elementary Number Theory, 2nd Ed. (date not set) New York, NY, Chelsea Publishing Co.,

Niven Ivan and Zuckerman, Herbert S., An Introduction to the Theory of Numbers, 4th edition, New York, NY, John Wiley and Sons, Inc., 1980. 2nd edition,

Petrofrezzo, A.J. and Byrkit, Donald, Elements of Number Theory, Winter Park, FL, Orange Publishers, 1985.

Weil, A., Basic Number Theory, 3rd edition, New York, NY, Springer Verlag, 1974.

Lectures, class discussions, assigned readings, problems and projects, written examinations.

At least sophomore status

Three semester hours

This course revision was examined in accord with established procedures and was approved by the Department of Mathematics on March 13, 1987.

___________________Chairperson

MAT 351 ELEMENTARY THEORY OF NUMBERS

Divisibility theory including Diophantine analysis, prime numbers and their distribution, Euler's phi function, theory of congruences, theorem of Fermat, Euler, and Wilson, primitive roots and indices.

Taught by members of the department having at least a Master's degree and preparation and interest in number theory.

A classroom with sufficient light, room for classroom activity and plenty of chalkboard space. The computer support currently available is probably sufficient.