MSM1303 - Number Theory

HNGU M.SC. MATHEMATICS SEMESTER - 3

MSM1303 – NUMBER THEORY

Unit 1: 

• Divisibility,
• G.C.D.,L.C.M
• Primes, 
• The fundamental theorem of arithmetic,
• The Euclidean algorithm,
• The greatest integer function,
• The Mobius function m,
• The Euler function F,
• The divisor functions sk for k³ 0 integer, properties of these functions,
• Multiplicative functions,
• Mobius inversion formula.

Unit 2: 

• Congruence,
• Complete residue systems,
• Linear Congruence,
• Reduced residue systems,
• Euler–Fermat theorem,
• The Chinese remainder theorem,
• The exponents of a number mod m,
• Primitive roots.

Unit 3:

• Quadratic Residues,
• Legendre Symbol and its properties,
• Gauss’ Lemma,
• The quadratic reciprocity law,
• The Jacobi Symbol.

Unit 4:

• Diophantine Equations ax + by = c and its positive solutions,
• the equation X2 + Y2 = Z2, the equation X4 + Y4 = Z2 and the equation X4 + Y4 = Z4,
• Sum of squares,
• The Fermat’s Last theorem.

Download Reference Books:

Note: The course is roughly covered by the book, Elementary Number Theory 2nd edition - David M. Burton (Wm. C. Brown Publishers, 1989).

1. An introduction to the theory of Numbers 3rd edition - I. Niven and H. Zukerman, Wiley Publication.

2. Introduction to Analytic Number Theorem - T. M. Apostol, Springer studt edition, 1995.

3. A concise Introduction to the theory of Numbers - Baker Alan, Cambridge University Press, 1984.

4. A course in number theory - Rose H. E., Oxford University Press, 1988.

5. Introduction to the theory of Numbers - Shapiro & Harold, John Wiley and Sons, 1983.

6. An Introduction to the theory of Numbers 5th edition - Hardy, G. H. and E. M. Wright, Oxford University Press, 1975.

7. Introduction to Number Theory 2nd edition - T. Nagell, chelsea, 1984. 

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