Math 453, Section X1: Exam 2 Study Guide
General Information
- Date/time/location:
The exam will be given during the regular class time,
Wednesday, 4/16/2008, 12:00 pm - 12:50 pm, in the regular
classroom, 145 Altgeld.
- Exam rules: Closed books and notes, and no calculators.
The problems will be such that they do not require significant
calculations if approached with the appropriate method.
Exam content
The exam will be on the material covered since the first midterm, i.e.,
Chapter 3 (arithmetic functions), Chapter 4 (quadratic residues),
and Chapter 5 (primitive roots), through Section 5.3.
(Section 5.4 will not be covered in class or in the hw,
so it won't be on the exam either.)
The following is a list of concepts, theorems, and techniques that you
need to be prepared for in the exam. If you are a bit fuzzy about a
particular item, review it from your class notes and from the appropriate
sections in the text; also redo any relevant hw problems.
Chapter 3: Arithmetic functions
- Notational conventions: Divisor sums and products, empty sum/product
convention
- Arithmetic function: Definition
- Multiplicative and completely multiplicative
arithmetic function: Definition and representation
in terms of prime factorization
- Dirichlet product of arithmetic functions: Definition and properties
(commutativity, associativity, identity element, Dirichlet inverse)
- Dirichlet product of multiplicative functions
- The three trivial arithmetic functions: delta function, unit function,
and identity function.
- The Moebius function (mu(n)): Definition and properties,
Moebius inversion formula
- The Euler phi function: Definition and properties, Gauss identity
- The Carmichael conjecture
- The number-of-divisors function: Definition and properties
- The sum-of-divisors function: Definition and properties
- Perfect numbers: Definition
- Perfect numbers: Characterization of even perfect numbers,
connection with Mersenne primes
Chapter 4: Quadratic residues
-
Quadratic residues and nonresidues: Definition
-
Number of quadratic residues and nonresidues modulo p
-
Number of solutions to quadratic congruences modulo p
-
Legendre symbol: Definition and properties (periodicity, complete
multiplicativity)
-
Legendre symbol: Value at -1, and at 2.
-
Euler's criterion for the Legendre symbol
(Note: You do not need to know Gauss' Lemma)
-
The Quadratic Reciprocity Law
Chapter 5: Primitive roots
-
The order of an integer: Definition and properties
-
The order of a power of an integer.
-
Number of elements of a given order modulo an odd prime p
-
Primitive root: Definition, and connection to reduced systems of
residues
-
The Primitive Root Theorem (Characterization of moduli for which a
primitive root exists)
-
Number of primitive roots modulo m
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Last modified: Sun 06 Apr 2008 05:23:42 PM CDT
A.J. Hildebrand