Math 117 What is Elementary Number theory Julius Magalona Basilla

Math 117 What is Elementary Number theory Julius Magalona Basilla

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Math 117-01
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Introduction
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Math 117
What is Elementary Number theory
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J.M.Basilla
Course
Information and
Policies
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Institute of Mathematics
University of the Philippines-Diliman
[email protected]
http://www.math.upd.edu.ph/faculty/jbasilla/
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Julius Magalona Basilla
19 September 2013
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The course
Math 117-01
J.M.Basilla
Course
Information and
Policies
Introduction
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course name
: Math 117 THV
course code
: 39358
course schedule : TTh 11:30-1:00P.M. MB 115
course instructor : J. M. Basilla
course homepage : http://www.math.upd.edu.ph/faculty/jbasilla/
Days Time
Venue
2:30-5:30 MB 229
consultation hours: TTh
WF
2:00-4:00 MB 229
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Course Requirements
Math 117-01
J.M.Basilla
Course
Information and
Policies
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1. 3 Long Exams (E1, E2,E3)
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Introduction
2. 1 Final Exam(F).
The final grade will be computed as follows
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2 E1 + E2 + E3 + max(E1, E2, E3) 1
+ F
3
4
3
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Final Grade =
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Transmutation Table
Math 117-01
J.M.Basilla
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Introduction
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Transmutted Grade
5.0
4.0
3.0
2.75
2.5
2.25
2.00
1.75
1.5
1.25
1.0
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Final Grade
0-54
55-59
59-64
65- 69
69-74
74 -79
80- 84
85-87
88-90
91-93
93-100
Course
Information and
Policies
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Class Policies
Math 117-01
J.M.Basilla
Course
Information and
Policies
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Introduction
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1. On Contention of grades. In case of a disagreement
with given grades, a student is given one week from the
date of receipt of marked examination papers(or
homeworks or project) or one week from the date of
posting in the class homepage, whichever applies.
2. Class Time. The class will be held from 11:35 A.M. to
12:50 P.M.
3. On Consultation. Consultation hours are reserved for
student asking clarificatory questions. It is not meant for
discussing lessons missed from absences.
4. On Missed Exam. All missed exam will initially merit
zero. Excused missed exam, with duly processed
papers, may be replaced by the lowest score among the
other two exams.
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Class Policies
Math 117-01
J.M.Basilla
Course
Information and
Policies
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Introduction
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5. On Cheating. Any form of cheating in examinations or
any act of dishonesty in relation to studies, such as
plagiarism, shall be subject to disciplinary action.
6. On Cellphone and other disruptive devices.
Cellphone and other disruptive devices are to be
turned-off or switch to silent mode during class time.
7. On Absences. Attendance will be checked. The
maximum number of unexcused absence is 6. Students
with more than 6 unexcused absences will be given a
grade of DRP only if the student has a passing class
standing. Otherwise, he/she will be given a grade of 5.0.
A student who misses the final exam will be given a
grade of 5.0 unless the student has a passing standing.
In this case, he/she will be given a grade of INC.
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Important Dates
Math 117-01
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J.M.Basilla
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Time
Class time
Class time
Class time
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7:00-9:00 A.M.
Event
Course
Information and
First Long Examination.
Policies
Second Long Examination
Introduction
Third Long Examination
Deadline for dropping
Final Exams
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Date
December 17, 2013
Feburary 11, 2014
March 20, 2014
February 20, 2014
March 31, 2014
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The number system
Math 117-01
J.M.Basilla
Course
Information and
Policies
R
Introduction
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N
Q
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Z
C
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1. N = 0, 1, 2, . . .(natural numbers).
2. The integers Z is the smallest set containing N such
that every equation X + a = b, a, b ∈ N has a solution.
3. The rationals Q is the smallest set containing Z such
that every equation aX = b, a 6= 0, b ∈ Z has a solution.
4. The reals, R is the set of limits of Caucy sequences
{ai } with rational terms ai ∈ Q.
5. The complex numbers C is the smallest set containing
R and the zeros of every polynomial f (X) ∈ R[X].
God made the integers; All else are man-made.
Leopold Krönecker
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What is Elementary Number Theory
Math 117-01
J.M.Basilla
Introduction
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Course
Information and
Policies
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i. Is a(a + 1)(a + 2) divisible by 6, for every integer a?
ii. Does the polynomial f (x) = x5 − x2 + x − 3 have integer
zeros?
iii. Consider the function
!
Y
1
, s>1
ζ(s) =
1 − p1s
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2. The third question is the primary focus in Elementary
Number Theory?
3. How difficult can these questions be?
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1. Given a complex number z ∈ C, is z real?
2. Given a real number r ∈ C, is r rational?
3. Given a rational number r ∈ Q, is r an integer?
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1. Common Mathematical questions
and the product
ζ(2) an integer?
Q
is taken over all prime numbers. Is
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