Index of the Lesson Plan 

UNIT – I NUMBER THEORY – 12 HRS 

L1  Introduction, Conjectures, Theorems and Proofs 
L2  Well ordering Principle, Principle of mathematical induction – strong and weak – problems 
L3  Primes and Composite numbers, Fibonacci numbers, Lucas numbers  problems 
L4  Sigma notation, Product notation, Binomial coefficients – properties, problems 
L5  Greatest integer function – properties, problems 
L6  Computers exercises and review exercises 
L7  Divisibility, proper divisor, nontrivial divisor, properties of divisibility, division algorithm 
L8  Common divisors, Greatest common divisor, properties of Greatest common divisor, Greatest common divisor via Euclid’s algorithm, Linear form of GCD, problems 
L9  Relatively prime numbers, Euclid’s lemma, pair wise relatively primes  problems 
L10  Common multiple, LCM, properties of LCM 
L11  Decimal representation of integers, binary representation of integers  problems 
L12  Computer exercises and review exercises 
UNIT – II PRIMES & CONGRUENCES – 12 HRS 

L13  Theorems on Composite and prime numbers, Twin primes 
L14  Prime counting function, Asymptotic functions, prime number theorem 
L15  Canonical factorization of a natural number, fundamental theorem of arithmetic 
L16  Dirichlet’s theorem, Bertrand’s postulates, Goldbach’s conjectures, problems 
L17  Test of Primality by trial division 
L18  Computer exercises and review exercises 
L19  Congruences and equivalence relations, equivalence class  problems 
L20  Least positive residue, linear congruences  problems 
L21  Chinese remainder theorem and problems 
L22  Modular arithmetic: Fermat’s little theorem 
L23  Fermat’s last theorem (Statement only), Wilson’s theorem and Fermat numbers 
L24  Computer exercises and review exercises 
UNIT – III ARITHMETIC FUNCTIONS, PRIMITIVE ROOTS 

L25  Arithmetic functions: sigma function, Tau function, Summation function, Sum of divisiors function, numbers of divisors function, problems 
L26  Multiplicative function, totally multiplicative function, Dirichlet product – properties and theorems, perfect numbers and Mersenne primes 
L27  Dirichelt inverse, Moebius function, Euler’s function, Euler’s theorem 
L28  Primitive roots – Definition and properties, Primitive root (mod m) 
L29  Index, Properties of indices – problems 
L30  Computer exercises and review exercises 
L31  Quadratic residue and nonresidue (mod p) 
L32  Euler’s criterion, properties of the Legendre symbol  problems 
L33  Gauss Lemma and related theorems 
L34  Law of quadratic reciprocity 
L35  Solution of quadratic congruences, Algorithm for solving quadratic congruences (mod p)  problems 
L36  Computer exercises and review exercises 
UNIT IV THE VECTOR SPACE R^{N}
& 

L37  Introduction to vectors, zero vector, Negative vector, Subtraction, Column Vectors 
L38  Dot Product, Norm, Angle and Distance, Norm of a Vector in R^{n} 
L39  Angle between Vectors, Distance between Points,
Introduction to Linear Transformations 
L40  Composition of Matrix Transformations. 
L41  Linear Transformations, Kernel and Range Terminology 
L42  Transformations and systems of Linear Equations,
Homogeneous Equations, Non homogeneous Equations 
L43  Many Systems, Coordinate Vectors, Notation, Change of Basis 
L44  Isomorphisms, Matrix Representations of Linear Transformations 
L45  Importance of Matrix Representations, Relations
between Matrix Representations 
L46  Diagonal Matrix Representation of a Linear Operator 
L47  Diagonalization of a matrixmatrix partitioning 
L48  Problems on L48 continued 
UNIT V NUMERICAL TECHNIQUES & 

L49  Gaussian Elimination, Comparison of GaussJordan
and Gaussian Elimination 
L50  Method of LU Decomposition, Construction of an LU decomposition of a Matrix 
L51  Practical Difficulties in solving Systems of
Equations, The Condition Number of a Matrix 
L52  Pivoting and Scaling Techniques 
L53  Iterative methods for solving systems of Linear
Equations, Jacobi Method 
L54  GaussSiedel Method 
L55  Eigen values by Iteration, Connectivity of Networks 
L56  Deflation Accessibility Index of a Network 
L57  A Geometrical Introduction to Linear Programming,
A Linear Programming Problem 
L58  Minimum Value of a Function, Discussion of the Method 
L59  The simplex Method, Geometrical Explanation of
the Simplex Method 
L60  Problems on L59 continued 
Test – I Unit I: Number Theory Unit II: Primes & Congruences 
Lesson Numbers L1 to L20 
Test – II Unit II: Primes & Congruences Unit III: Arithmetic Functions, Primitive roots & Indices & Quadratic Congruences 
L21 to L36 
Test – III Unit IV: The vector space and Linear Transformation Unit V: Numerical Techniques & Linear Programming 
L37 to L60 